I was going through the rulebook and cannot find a mention of an exact date or x number of days. Here is a link to the online rulebook - AMHA Rule Book
I believe right now all foals that are DNA tested are automatically PQ'd after 2005 if both parents have been DNA'd. I know after the 1/1/1996 'must be DNA'd' date AMHA can have a horse PQ tested without the owners permission. But again can't find an exact reference to your question . . . hmmm.
As a rule I get all our foals DNA/PQ tested so I've never been concerned about a time frame.
How many days late can a foal be before AMHA requires it to be Parent Qualified?