They were illegimate and didn't enjoy the same rights as full blooded spaniards for sure. But the illegimate male children of spaniards had better mating privileges as nearly two thirds of mexican males have european Y chromosomes. That can't happen without the illegimate male offprings of spaniards having privileges that allowed them to mate at a much higher rate than native males since the full blooded spaniard population was never great enough to produce such drastic genetic disparity. What the privileges and rights were, who knows?
Do you have any sources on that? As far as I know they had the same rights given by the Crown (legally at least).
Legitimate mestizos enjoyed privileges. Otherwise they had to stay close to their parents.
But native women could not represent themselves in court so they could not challenge a Spanish man to recognize their children.