Calling Animation in 2 Different Widgets from a 3rd (Not the Parent Widget)

All you need to do is to pass the AnimationController from respective BodyPage to your PrimaryMenu which is responsible to call Page1 and Page2. Once the AnimationController is available to PrimaryMenu it will reverse the animation first and .then call the respective page. Please see the working code below :

import 'package:flutter/material.dart';

void main() {
  runApp(MyApp());
}

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      theme: ThemeData(visualDensity: VisualDensity.adaptivePlatformDensity),
      debugShowCheckedModeBanner: false,
      home: Page1(),
    );
  }
}

class Header extends StatefulWidget {
  @override
  _HeaderState createState() => _HeaderState();
}

class _HeaderState extends State<Header> with SingleTickerProviderStateMixin {
  AnimationController transitionAnimation;

  @override
  void initState() {
    super.initState();
    transitionAnimation = AnimationController(
      duration: const Duration(seconds: 3),
      vsync: this,
    );
    transitionAnimation.forward();
  }

  @override
  Widget build(BuildContext context) {
    return Container(
        color: const Color(0x88dddddd),
        height: MediaQuery.of(context).size.height,
        width: MediaQuery.of(context).size.width,
        child: Row(
          crossAxisAlignment: CrossAxisAlignment.center,
          children: [
            const SizedBox(
              width: 8,
            ),
            AnimatedBuilder(
              animation: transitionAnimation,
              builder: (context, child) {
                return SlideTransition(
                  position: Tween<Offset>(
                          begin: const Offset(-2, 0), end: const Offset(0, 0))
                      .animate(CurvedAnimation(
                          curve: const Interval(0, 0.3,
                              curve: Curves.easeInOutBack),
                          parent: transitionAnimation)),
                  child: child,
                );
              },
              child: Container(
                height: 100,
                width: MediaQuery.of(context).size.width * 0.135,
                color: Colors.pink,
              ),
            ),
            Expanded(
              child: Container(),
            ),
            Container(
              width: 100,
              height: 50,
              color: Colors.purple,
            ),
            const SizedBox(
              width: 8,
            ),
            Container(
              height: 50,
              width: 50,
              color: Colors.amber,
            ),
            const SizedBox(
              width: 8,
            )
          ],
        ));
  }
}

class Page1 extends StatefulWidget {
  @override
  _Page1State createState() => _Page1State();
}

class _Page1State extends State<Page1> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      extendBodyBehindAppBar: true,
      appBar: PreferredSize(
        preferredSize: Size(MediaQuery.of(context).size.width, 120),
        child: Header(),
      ),
      body: BodyPage1(),
    );
  }
}

class PrimaryMenu extends StatefulWidget {
  final AnimationController controller;

  const PrimaryMenu({Key key, this.controller}) : super(key: key);
  @override
  _PrimaryMenuState createState() => _PrimaryMenuState();
}

class _PrimaryMenuState extends State<PrimaryMenu> {
  @override
  Widget build(BuildContext context) {
    return Container(
        height: MediaQuery.of(context).size.height,
        width: MediaQuery.of(context).size.width * 0.15,
        color: Colors.blue,
        child: Column(
          mainAxisAlignment: MainAxisAlignment.center,
          children: [
            Container(
                height: 40,
                width: MediaQuery.of(context).size.width * 0.135,
                color: Colors.grey,
                child: InkWell(
                  onTap: () async {
                    print('1');
                    await widget.controller
                        .reverse()
                        .then((value) => Navigator.push(
                            context,
                            MaterialPageRoute(
                              builder: (context) => Page1(),
                            )));
                  },
                )),
            const SizedBox(height: 16),
            Container(
                height: 40,
                width: MediaQuery.of(context).size.width * 0.135,
                color: Colors.black,
                child: InkWell(
                  onTap: () async {
                    print('2');
                    await widget.controller
                        .reverse()
                        .then((value) => Navigator.push(
                            context,
                            MaterialPageRoute(
                              builder: (context) => Page2(),
                            )));
                  },
                )),
          ],
        ));
  }
}

class BodyPage1 extends StatefulWidget {
  @override
  _BodyPage1State createState() => _BodyPage1State();
}

class _BodyPage1State extends State<BodyPage1>
    with SingleTickerProviderStateMixin {
  AnimationController transitionAnimation;

  @override
  void initState() {
    super.initState();
    transitionAnimation = AnimationController(
      duration: const Duration(seconds: 3),
      vsync: this,
    );
    transitionAnimation.forward();
  }

  @override
  Widget build(BuildContext context) {
    return Row(
      children: [
        AnimatedBuilder(
            animation: transitionAnimation,
            builder: (context, child) {
              return SlideTransition(
                position: Tween<Offset>(
                        begin: const Offset(-2, 0), end: const Offset(0, 0))
                    .animate(CurvedAnimation(
                        curve:
                            const Interval(0, 0.3, curve: Curves.easeInOutBack),
                        parent: transitionAnimation)),
                child: child,
              );
            },
            child: PrimaryMenu(
              controller: transitionAnimation,
            )),
        AnimatedBuilder(
          animation: transitionAnimation,
          builder: (context, child) {
            return SlideTransition(
                position: Tween<Offset>(
                        begin: const Offset(-2, 0), end: const Offset(0, 0))
                    .animate(CurvedAnimation(
                        curve: const Interval(0.3, 1, curve: Curves.easeIn),
                        parent: transitionAnimation)),
                child: child);
          },
          child: Padding(
            padding: const EdgeInsets.only(
                top: 140, left: 20, right: 20, bottom: 20),
            child: Container(
              height: MediaQuery.of(context).size.height - 140,
              width: (MediaQuery.of(context).size.width * .85) - 40,
              color: Colors.grey,
            ),
          ),
        )
      ],
    );
  }
}

class Page2 extends StatefulWidget {
  @override
  _Page2State createState() => _Page2State();
}

class _Page2State extends State<Page2> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      extendBodyBehindAppBar: true,
      appBar: PreferredSize(
        preferredSize: Size(MediaQuery.of(context).size.width, 120),
        child: Header(),
      ),
      body: BodyPage2(),
    );
  }
}

class BodyPage2 extends StatefulWidget {
  @override
  _BodyPage2State createState() => _BodyPage2State();
}

class _BodyPage2State extends State<BodyPage2>
    with SingleTickerProviderStateMixin {
  AnimationController transitionAnimation;

  @override
  void initState() {
    super.initState();
    transitionAnimation = AnimationController(
      duration: const Duration(seconds: 3),
      vsync: this,
    );
    transitionAnimation.forward();
  }

  @override
  Widget build(BuildContext context) {
    return Row(
      children: [
        AnimatedBuilder(
            animation: transitionAnimation,
            builder: (context, child) {
              return SlideTransition(
                position: Tween<Offset>(
                        begin: const Offset(-2, 0), end: const Offset(0, 0))
                    .animate(CurvedAnimation(
                        curve:
                            const Interval(0, 0.3, curve: Curves.easeInOutBack),
                        parent: transitionAnimation)),
                child: child,
              );
            },
            child: PrimaryMenu(
              controller: transitionAnimation,
            )),
        AnimatedBuilder(
          animation: transitionAnimation,
          builder: (context, child) {
            return SlideTransition(
                position: Tween<Offset>(
                        begin: const Offset(-2, 0), end: const Offset(0, 0))
                    .animate(CurvedAnimation(
                        curve: const Interval(0.3, 1, curve: Curves.easeIn),
                        parent: transitionAnimation)),
                child: child);
          },
          child: Padding(
            padding: const EdgeInsets.only(
                top: 140, left: 20, right: 20, bottom: 20),
            child: Container(
              height: MediaQuery.of(context).size.height - 140,
              width: (MediaQuery.of(context).size.width * .85) - 40,
              color: Colors.black,
            ),
          ),
        )
      ],
    );
  }
}