In the wedding scene (prior to his family being taken into witness protection) we learn that his name is Robert Parr.
Minister: Robert Parr, will you have this woman to be your lawful wedded wife?
and it was Robert "Bob" Parr when he met Elastigirl in the first place.
"Oh, well...I thought I'd see you there." He let out a chuckle as he
shook his head. "Whoops, I never even introduced myself! I'm Bob
Parr." He wiggled his fingers, which were still wrapped around his
shopping bags. "I'd shake, but..." "Helen," Elastigirl said. She
shifted her own grocery bag so she could take one of his fingers. She
shook it. "Nice to meet you."
A Real Stretch: An Elastigirl Prequel Story
It seems likely that when a new identity is created, it's created in their same names, noting that there are literally hundreds of people named Robert Parr in the US.
On a related note, Helen Parr's maiden name was supposedly Truax, according to the film's creator Brad Bird.
Q. Is Helen Parr's maiden name really Truax?
Bird: Uh... yes (a detail that hasn't been in my mind for a long time).
Interestingly, in A Real Stretch: An Elastigirl Prequel Story we learn that she goes by at least one other name, Helen Highwater.
Then she left to greet the world as Helen Highwater, Elastigirl's secret identity. Most Supers had them, and Elastigirl found hers immensely helpful. After all, it would be pretty hard to go about your day doing chores and running errands as a Super. People would stop you for an autograph, or to take a picture, or to ask you to do things for them that didn't truly qualify for a Super's intervention.
Note that at this point she's working directly for the NSA so there's every possibility that this surname is just a cover. She does refer to herself in internal dialogue as Helen, so that's probably her real name.