What can be the consequences with the law for a 20 year old male dating a 16 year old female with parental approval? He is also my first cousin so would that help us in anyway because we are family?

Oh yeah, sure, being family will most certainly help... (sarcasm)

So where are you located?

If you're having sex, it's likely that he's going to jail as a sex offender at some point.

If this is the US, being first cousins make it a lot worst, esp if you are having sex.

Since there is no age of consent for 1st cousins. There is no "dating" law but there are several laws about who you can have sex with. So where you live will make a difference.

In US, a parent can not give permission for illegal activities.