I think this has been discussed before, but can't remember or find it in the online rule book..which seems to be the 2005 edition. Not sure if there have been updates wrt this issue?
Your answer is correct but may I clarify just a bit.
A stallion can sire a foal at any age but the offspring cannot be registered unless the dam and sire are 3 years of age (as of January 1) during the year of foaling
This is exactly how the rule book reads.
No Miniature foal may be registered out of a sire or dam unless BOTH have reached their third birthday (birthday as of January 1) during the foaling year.
So say a stallion was born Dec 31 2003. On January 1 he becomes a yearling at 1 day old. In essence he could breed a mare when he was 14 months old and have a registerable foal. I really think this rule needs to be reworded and go back to actual birthdate of sire and dam.