I'm not completely sure what you want (a better description would be great), but what I understand from your answer you could do something like this:

@Component({
  selector: 'app-component',
  template: `
    <button (click)="toggleDarkMode()">Toggle dark mode</button>
    <p>{{isDarkMode ? 'dark' : 'light'}} mode</p>
  `
})
export class AppComponent {
  public isDarkMode = false;

  public toggleDarkMode(): void {
    this.isDarkMode = !this.isDarkMode;
  }
}

or:

type Theme = 'dark' | 'light';

@Component({
  selector: 'app-component',
  template: `
    <button (click)="toggleDarkMode()">Toggle dark mode</button>
    <p>{{theme}} mode</p>
  `
})
export class AppComponent {
  public theme: Theme = 'light';

  public toggleDarkMode(): void {
    this.theme = this.theme === 'light' ? 'dark' : 'light';
  }
}

Edit

In the darkmode-toggler component you can do what was mentioned above to change the paragraph's text.

And then use @Output to give it a custom event.

If I edit your sample it would look like this:

@Component({
  selector: 'darkmode-toggler',
  template: `
    <div>
      <button>Toggle dark mode</button>
      <p>{{isDarkMode ? 'dark' : 'light'}} mode</p>
    </div>
  `,
  styles: []
})
export class DarkModeToggler {
  @Output() stateChanged = new EventEmitter<boolean>();
  isDarkMode = false;

  public toggleDarkMode(): void {
    this.isDarkMode = !this.isDarkMode;
    this.stateChanged.emit(this.isDarkMode); // you emit a change here
  }
}

@Component({
  selector: 'app-root',
  template: `
    <div>
      <darkmode-toggler (stateChanged)="handleStateChange($event)"> 
      </darkmode-toggler>
    </div>
  `,
  styles: []
})
export class Resource {  
  handleStateChange(event) {
    console.log(event); // since I made a EventEmitter<boolean>() and I'm emitting the isDarkMode variable's value it'll be true or false
  }
}