I don't really have that problem so much getting caught somewhat off guard when somebody greets me or says goodbye, where I'll usually end up awkwardly mumbling something in response, often mirroring what they said, which isn't so bad when it's just "Hi," but responding to a "Nice to meet you," with another "Nice to meet you," is a little bit strange.
Out of curiosity, are these "conversation mantras" something that you'd like change? If not, then please feel free to ignore me, but if so, then perhaps you could try to enter conversations with your dad already planning what you're going to say after you greet each other, so that you don't end up falling back on your default of "Hi." Ask him how his day has gone, or bring up some subject that interests both of you, or I'm sure you can think of something.