Complexity of counting when there is no parsimonious reduction
I don't think that having a non-parsimonious reduction helps in proving that a counting problem is in FP. Eg. if either 1 or 2 instances are mapped to 1, and which of these depends on something NP-hard, then you do not know anything about the complexity of your problem. Of course, you might have a special reduction from which you get some extra information.