Gabriel Aubry is really shouldering the burden of fatherhood -- giving his 4-year-old daughter Nahla an adorable piggyback ride in L.A. yesterday ... after picking her up from school. 

And there's plenty more where that came from -- because as we reported, Gabriel will NOT be criminally prosecuted for his involvement in a Thanksgiving Day brawl with Halle Berry's fiance Olivier Martinez ... as long as he keeps his nose clean for a year.

Worth noting ... neither Halle nor Olivier wanted to press charges against Gabriel over the fight because they didn't think it would be in Nahla's best interests.