I slip between Mother, Mom, and Momma when I talk to my mother, depending entirely on the conversation. Generally I use "Mother" when I'm irritated or pretending I'm irritated, like parents use middle names. I use Mom and Momma interchangeably, although if I'm feeling puny or sad I probably use Momma more.
I use Dad and Daddy equally, and rarely (if ever) refer to my father as Father.
If I'm talking to someone else about them, though, I will like say "My mother," or "My father," or sub in Mom and Dad. I see it as varying levels of formality in addressing them, if that makes any sense.
It depends on your protagonist, but I think that if you establish early on that they use both, then it shouldn't be an issue. Everyone's different.