Yes you can direct link to a page.
I’m not sure what the issue is that you are having.
But you can go to “my homepage” then go to “another page on my website” and the embed will work
Since the parent is jsut a list of domains (including sub domain) (never contains path so you parent is www.example.com not www.example.com/twitch
Sounds like you may need to include both the www. and non www. versions in your parent
So
https://player.twitch.tv/?channel=xxxxxx&parent=example.org&parent=www.example.org
To use your example
As you might be landing on example.org then going to www.example.org/twitch