What's the relationship between ASP-complete and #P-complete? Given that ASP-reductions by definition are parsimonious and parsimonious reductions preserve #P-completeness, one might think that the counting version of all ASP-complete problems are also #P-complete (note that e.g. #3SAT is #P-complete and 3SAT is ASP-complete).

However, ASP-completeness implies that all solutions have polynomial length, whereas no such requirement is there for #P.

Is it the case that all ASP-complete problems are #P-complete?  Is there some kind of relationship in the other direction as well?
Let an ASP-complete problem Q be given.  There is an ASP-reduction from 3SAT to Q and all ASP-reductions are parsimonious, hence it serves as a parsimonious reduction from #3SAT to #Q.  Since #Q is in #P and #3SAT is #P-hard, #Q is #P-complete.