Can a 21-year-old get into legal trouble if he gets a 16-year-old pregnant if her mother agreed to the relationship?
Yes, how much will depend on the laws of the specific state. And the mother's 'permission' doesn't negate anything, in fact can get her in trouble for contributing to the illegal act.
No. When parents are uncomfortable with their child's relationship with a 21+ (with the child being under 21) they can get sued for being a sex offender (only of course they did have a sexual relationship) then of course you are very in trouble with the law. But until your 21 and she's 19 you might get in trouble.