Being a Paratroopa myself, I remember exactly how it went down. During SMB, SMB3, and any other game featuring the Troopas on all fours, there was a reward going around for anyone who could find Bowser's lost contact lens. Hence, the reason Mario was able to beat him was that he lacked depth perception. Eventually, he got another pair, so we stopped looking.
"A person is a person, no matter how small."- Horton the Elephant