Is the autistic child over age 18 or under?
If under, then it would depend whether the child's autism qualifies as a "serious health condition" as defined in the FMLA. (See 29 CFR 825.114). If the child is over age 18, then the parents' eligibility also depends on whether the child is capable of self-care.
There are very mild forms of autism to quite severe. Just because someone is autistic doesn't automatically mean it is a serious or disabling health condition.
A person, who is nice to you, but rude to a waiter, is not a nice person. (This is very important. Pay attention. It never fails.)