What is the complexity of counting parse trees?
Because you are asking the question in general, the answer will be: uncertain number of trees. 

Consider a degenerate case, when you have such a CFG:

$S\rightarrow aA$

$A\rightarrow B$

$B\rightarrow A$

$A\rightarrow aS$

$S\rightarrow a$

This is a valid grammar, right? Though not normalized.
Obviously you can "run" between $A$ and $B$ for any number of times for any given string $s$ generated by this grammar. Each transition between $A$ and $B$ gives you new node in a parse tree, thus we have uncertain number of them.

So, I would rather ask about normalised CFG. For a normalised CFG I guess it should be only one parse tree per the given string $s$.