Hi, I was wondering if a 15 day land or 30 day air, visa exempt entry can still be extended for 60 days based on supporting a child? Thanks.

Yes, but how many times depends on the immigrations office.

Once per year, once per entry, no specified rule.

From Police order 777/2551

2.24 In the case of an alien visiting a Thai spouse or child:

Permission will be granted for a period of not more than 60 days at a time.

(1) Proof of relationship.

(2) In the case of a spouse, their marital relationship shall be de jure (legitimate) and de facto.

Thanks for that, I'll only have 1 free page left after they stick a visa in the passport, will this be ok to enter on (then get a new passport at the embassy in BKK)?

It is not a visa and will take less than half page so should be fine.