I don't think that thread suggested that horses can have Down's Syndrome, but perhaps I misunderstood it?
I read it as saying that the age of the mare's eggs does not lead to genetic defects as such, rather there may be more chance, with time, that a defective egg becomes fertilized. The defects referred to were general defects, not Down's Syndrome.
The thread also moved on to consider genetic abnormalities in humans and the likelihood of older women giving birth to children with Down's.