As Tamera said, if I were talking about myself, I would likely say "my mom," or "my dad."
But if I'm talking to a child and I want to refer to one of the adults responsible for them (but I don't care which one), I may use "parent." e.g., "Where is your parent?" "Make sure that your parent brings you here at 8 tomorrow morning." "Did your parent give you permission to do that?"