“Health” here is being understood very narrowly.
Given the mechanism of male reproduction (copies of copies of copies being made) it is very natural that errors arise in the copying process, and more and more as time goes by. Hence, it is now increasingly understood that paternal age does factor into chromosomal abnormalities, as paternal age is a source of abnormalities via mutation.
With autism, paternal age seems to have a stronger effect than maternal age:
medscape.com/viewarticle/846265
"The influence of paternal age on the occurrence of genetic disorders, including Down syndrome, has long been thought to be secondary. However, the influence of older paternal age at childbearing should not be ignored. New mutations leading to conditions such as neurofibromatosis and achondroplasia (a form of dwarfism) are more frequent in older men and therefore are more common in the children of older fathers.
“In addition, Fisch and colleagues studied the influence of paternal age on Down syndrome and showed that “advanced paternal age combined with maternal age significantly influences the incidence of Down syndrome.””
medscape.com/viewarticle/496962
And this area of study is fairly new, so this is just the beginning of our knowledge about the effect of advancing paternal age on chromosomal abnormalities. Within the next 10-20 years, I fully expect that we will be able to add a lot more items to the list of risks run by the children of older fathers.