The question as phrased, the answer is no. It is not legal. If the 16 year old engages in any sexual conduct with the 13 year old, they could face statutory rape charges and the parental consent assuming there was any would have no bearing
Is it bad for a 15 year old to date a 13 year old?
If dating includes sex (or any form of sexual activity) then the answer is NO - and parental approval is irrelevant. If you are under 16 then you may not lawfully engage in any sexual activity (not just intercourse) with anyone.
Is it OK for a 13 year old to date a 17 year old?
No. It is not illegal for a 13 and 17 year old to date.