Solution

This method worked for me (I've modified some things slightly to post in a public forum) EDIT: I have updated this to use the Blue/Green solution now that I have had the chance to test it.

Parameters:
[...]
  DBEngineVersion:
    Type: String
    Description: "Choices are engine versions from Postgres 9/10/11, only newest and legacy deployed in Prod versions"
    Default: 11.7
    AllowedValues:
      - 9.6.18
      - 10.7
      - 10.14
      - 11.7
  MajorVersionUpgrade:
    Type: String
    Description: Swap this between 'Blue' or 'Green' if we are doing a Major version upgrade
    Default: Blue
    AllowedValues:
      - Blue
      - Green

Conditions:
  BlueDeployment: !Equals [!Ref MajorVersionUpgrade, "Blue"]
  GreenDeployment: !Equals [!Ref MajorVersionUpgrade, "Green"]
  
Mappings:
  EngineToPGFamily:
    "9.6.18":
      PGFamily: aurora-postgresql9.6
    "10.7":
      PGFamily: aurora-postgresql10
    "10.14":
      PGFamily: aurora-postgresql10
    "11.7":
      PGFamily: aurora-postgresql11

  ##### Keep parameters of both versions in sync ####
  DBClusterParameterGroupBlue: #Keep parameters of both versions in sync
    Type: "AWS::RDS::DBClusterParameterGroup"
    Condition: BlueDeployment
    Properties:
      Description: !Sub "Postgres Cluster Parameter Group for ${InfrastructureStackName}"
      Family: !FindInMap ['EngineToPGFamily', !Ref 'DBEngineVersion', 'PGFamily']
      Parameters:
        client_encoding: UTF8
        idle_in_transaction_session_timeout: 60000
  DBClusterParameterGroupGreen:
    Type: "AWS::RDS::DBClusterParameterGroup"
    Condition: GreenDeployment
    Properties:
      Description: !Sub "Postgres Cluster Parameter Group for ${InfrastructureStackName}"
      Family: !FindInMap ['EngineToPGFamily', !Ref 'DBEngineVersion', 'PGFamily']
      Parameters:
        client_encoding: UTF8
        idle_in_transaction_session_timeout: 60000

  DBCluster:
    Type: "AWS::RDS::DBCluster"
    Properties:
      DBClusterIdentifier: !Sub "${AWS::StackName}-cluster"
      DBClusterParameterGroupName: !If [GreenDeployment, !Ref DBClusterParameterGroupGreen, !Ref DBClusterParameterGroupBlue]
      Engine: aurora-postgresql
[...]
  DBInstance1:
    Type: "AWS::RDS::DBInstance"
    Properties:
      DBClusterIdentifier: !Ref DBCluster
      EngineVersion: !Ref DBEngineVersion
      Engine: aurora-postgresql
[...]

  DBInstance2:
    Type: "AWS::RDS::DBInstance"
    Properties:
      DBClusterIdentifier: !Ref DBCluster
      EngineVersion: !Ref DBEngineVersion
      Engine: aurora-postgresql
[...]

Usage and Methodology

This method will create a new ClusterParameterGroup that will be used during the upgrade (when doing a Major version upgrade), and clean up the original group when the upgrade is done. The advantage to this solution is that there is no version hardcoding in the group and the same template can be used for continuous lifecycle updates without additional changes needed (aside from desired Parameters updates).

Initial Setup

Initially, leave MajorVersionUpgrade set to Blue and the conditions will ensure that only the Blue ParameterGroup is created using the proper DB family.

Performing a Major Version Upgrade

Update the stack and set the DBEngineVersion to the new major version and MajorVersionUpgrade to Green if it is currently Blue (or Blue if it is currently Green). CloudFormation will:

1. Start by creating the new resources (which is the new parameter group referencing the new version)
2. Then it will update the Cluster and point to that new group
3. Finally, it will delete the old ParameterGroup (referncing the old version) during the cleanup phase since the Blue/Green Condition for that group no longer evaluates to true.

Important notes

1. The Parameters in both parameter groups must manually kept in sync
2. Obviously, update the DBEngineVersion AllowedValues list with versions as needed, just be sure to update the EngineToPGFamily Mappings to have a valid Postgres family value for that Engine version
3. Special care must be paid when implementing Parameters that are new to a DB Major version and don't exist in an older version, I have not tested this

You are correct that you cannot change the family on a DB Parameter Group that already has been created. That is why you get the error about the missing parameters, which isn't a great error response by AWS given the root cause.

Since DB Parameter Groups can be replaced without having to replace the DB Instance, you should change the Logical ID of the DB Parameter Group so it is detected as a new resource by CloudFormation, and removes the old v10 DB Parameter Group.

Example using your code:

Parameters:

  pEngineVersion:
    Type: String
    Default: '11.9'
    #currently '10.14'
    #trying to change to '11.9'
  pFamily:
    Type: String
    Default: 'aurora-postgresql11'
    #currently 'aurora-postgresql10'
    #trying to change to 'aurora-postgresql11'

Resources:

  DBClusterParamGroup11:
    Type: AWS::RDS::DBClusterParameterGroup
    Properties:
      Description: !Sub 'AuroraDBClusterParamGroup-${AWS::Region}'
      Family: !Ref pFamily
      Parameters:
        application_name: "App name"
        log_statement: all
        log_min_duration_statement: 0

  DBParamGroup11:
    Type: AWS::RDS::DBParameterGroup
    Properties:
      Description: !Sub 'AuroraDBParamGroup-${AWS::Region}'
      Family: !Ref pFamily
      Parameters:
        max_connections: 1000

  AuroraDBCluster:
    Type: AWS::RDS::DBCluster
    Properties:
      EngineVersion: !Ref pEngineVersion
      Engine: aurora-postgresql
      DBClusterParameterGroupName: !Ref 'DBClusterParamGroup'
      #snipping unneccesary code#

  AuroraDBInstance:
    Type: AWS::RDS::DBInstance
    Properties:
      Engine: aurora-postgresql
      DBParameterGroupName: !Ref 'DBParamGroup'
      DBClusterIdentifier: !Ref 'AuroraDBCluster'
      #snipping unneccesary code#

Notice how I renamed DBParamGroup to DBParamGroup11 and DBClusterParamGroup to DBClusterParamGroup11.