I'm revisiting this exercise now and AGAIN not sure about this particular thing :)
"I like cooking with dad" - sounds fine to me.
"I like cooking with father" - sounds a liiiiittle bit awkward...
Do we have native speakers here? Can anyone comment? I'll ask my friends IRL...
I'm a native speaker of US English. "Dad", "Mom" etc. without a possessive are used by family members when speaking within the family to refer to the parents.
Example: "Susan (referring to a family member) loves cooking with Dad", or "She loves cooking with Dad". You could say "She loves cooking with her Dad", but it's a bit ambiguous since it could mean the father of another female who isn't a member of that family.
If speaking to someone outside the family, you would almost always use the possessive, unless the person you are speaking with is so close to the family that the possessive would sound too formal.
In any case, for this sentence, with no context, I believe that "with Dad" should be accepted and I've reported it.
If you really do want to emphasize, you can add "свій": Він любить готувати зі своїм батьком. If you write "Він любить готувати з його батьком" it's still perfectly correct, but it means with someone else's father, there's another "він" in the context.
With the first and second person it's simply wrong: Я люблю свою машину (NOT я люблю мою машину)
With the third person it's two different meanings: Він любить свою машину (his own car). Він любить його машину (someone's car)
Please specify what you wrote as an answer to this exercise so that we can see why it wasn't accepted :)
Or if you want to ask more about the sentence with "mother with his father" that you mentioned, please also write your input to that, because I don't really understand literal or non-literal translation of what you're talking about...