Consistent query answering (CQA) aims to deliver meaningful answers when queries are evaluated over inconsistent databases. Such answers must be certainly true in all repairs, which are consistent databases whose difference from the inconsistent one is somehow minimal. An interesting task in this context is to count the number of repairs that entail the query. This problem has been already studied for conjunctive queries and primary keys; we know that it is #P-complete in data complexity under polynomial-time Turing reductions (a.k.a. Cook reductions). However, as it has been already observed in the literature of counting complexity, there are problems that are 'hard-to-count-easy-to-decide', which cannot be complete (under reasonable assumptions) for #P under weaker reductions, and, in particular, under standard many-one logspace reductions (a.k.a. parsimonious reductions). For such 'hard-to-count-easy-to-decide' problems, a crucial question is whether we can determine their exact complexity by looking for subclasses of #P to which they belong. Ideally, we would like to show that such a problem is complete for a subclass of #P under many-one logspace reductions. The main goal of this work is to perform such a refined analysis for the problem of counting the number of repairs under primary keys that entail the query.