I put "My father bakes cake" instead of "a cake." This was also accepted, but I'm wondering if there's a way to make this distinction. The first is more like saying "one of the things my father can do is bake cakes" (or maybe as the answer to "what does your father bake?") while the second might also be translated as "my father is baking a cake" meaning "that's what he's doing right now."
Why don't אבא and אמא have the "ה" in front for posession? For example, I would use "הכלב שלי" for my dog, but it wants me to use "אבא שלי" for my father. And are there other cases where the "ה" is dropped?
אבא and אמא is always definite because you have usually only one father and one mother, so you do not need ה. I think proper names are also always definite.
The uniqueness of father and mother is surely an important part of the reason. But it's not a sufficient condition. There are other things of which you usually only have one, and (linguistically) they are not definite by themselves in Hebrew: חתן, כלה, בעל, אישה, בית, ארץ, מדינה, עיר, מקום עבודה.
So I think the full answer to Paige's question is "because father and mother are typically unique, and because."