When I answered Yes, I have mom and dad it marked me wrong for omitting the article "a" before "mom". It didn't require me to write "a dad" though. Wouldn't it make more sense if there was either "I have mom and dad" or "I have a mom and a dad" but not only one of them "a mom" but "dad", not "a dad"?
To me, it depends on the focus. "A mom and a dad" focuses on both parents as individuals and "a mom and dad" is almost equivalent to "parents", where the focus is on the two as a collective unit. I might use "a mom and a dad" to explain to a child where babies come from, for example.
Does 'У' means 'by' or 'in/inside'? According to sentence 'Я у кафе' (inside cafe) this preposition is different from 'y' in Russian (in Russian this sentence would mean - I am by the entrance to cafe'. But here 'y' is used as well as in Russian 'I have parents by my side' (not 'They are inside me in my stomach')
Ukrainian "у" is different from Russian one. Here it's just an equivalent to "в" and means "in".
If you were to say 'They are inside me in my stomach', you'd need dative "у мені"
"У мене є" is rather idiomatic and means "I have". You use just "у мене" (without є) to speaks about diseases: У мене грип (I have the flu). Also colloquially people say У мене when you'd expect them to say "my": У мене сестра вчиться за кордоном (My sister studies abroad).