It’s hard to say. I doubt Diana would ever call Nick “dad.” Granted, she might if she grew up being parented by both him and Sean more or less equally from a young age.
I grew up in a very blended family situation, and I’ve seen both. I call my stepmother “momma,” for example, because she’s been a mother to me since I was three. Then again, one of my sisters calls her parents “mom” and “dad” and my father “Uncle D—.” And my other sister calls her parents “mom” and “dad” and her sister’s father by his first name. And it’s just understood that whoever’s talking means their biological parents when they say “mom” or “dad.”
So it’s entirely possible that when she says “mom and dad” she means Adalind and Sean, and Kelly understands this. It’s also possible that at some point she started calling Nick “dad” as well…but I find it somewhat unlikely, since she consistently called him Nick throughout the series.