Types of reductions and associated definitions of hardness
Your definition of parsimonious reductions is incorrect. You are confusing it with polynomial-time one-one reductions which is a special case of Karp reductions. They don't preserve the number of "solutions". Please see this answer for more about reductions considering the number of certificates.

The rest seems fine although it is usually better to view them in a two dimensional chart:  


the complexity of reduction: computable, polynomial time, logarithmic space, etc.
type of access: Turing, many-one, one-one, etc. 



  Are NP-complete problems defined with respect Cook reduction or parsimonious reduction? 


$\mathsf{NP}$ hardness and completeness are defined w.r.t. Karp reductions (polytime many-one), not Cook nor parsimonious reductions.


  Can anyone kindly give an example of a problem that is NP-complete under Cook and not under parsimonious reduction.


Take the complement of SAT, it is complete for $\mathsf{NP}$ under Cook reductions, it is not believed to be complete for $\mathsf{NP}$ under Karp reductions. Karp reductions include polytime one-one reductions.


  the class #P-Complete is defined with respect to Karp reductions


Note that $\mathsf{\#P}$ is not class of decisions problems, it is a class of function computation problems. Its hardness and completeness are usually defined w.r.t. Cook reductions (polytime Turing). See e.g. Arora and Barak, page 346.