Parsons put the Cowboys in command in the third quarter of their 49-29 win at AT&T Stadium on Sunday when he returned a David Montgomery fumble 36 yards for the score.
Parsons got back in position to recover the ball and, after realising Fields had leapt over him rather than touch him down by contact, evaded several scrambling Bears players to find the endzone and give the Cowboys a 42-23 lead.
It marked Parsons' first touchdown of his NFL career, and he now feels he should have the chance to add to that tally on offense.
"Zeke [Ezekiel Elliott] said I look just like him," Parsons said of his touchdown.
"I think I definitely deserve a rep now. They can't say they haven't seen it in a game. So hopefully I get thrown in a goal-line package or red-zone package."
Cowboys head coach Mike McCarthy, whose team are 6-2 behind only the 7-0 Philadelphia Eagles in the NFC East, is concerned the score will lead to him being pestered by Parsons to play on offense.
"I was obviously very excited," McCarthy said. "But part of me is annoyed because he wants to be on offense already.
"Now I'm going to have a hard time keeping him out of my office."