Is conversion of PRAM to parameter number of processors trivial
here are my answers to your questions.

1) In general, you can not come up with efficient schemes for scaling down the number of processor from $n$ to $p &lt; n$. Why ? Because, even without taking into account additional issues such as caches etc, the problem is strictly related to how you map tasks to processors, which is known to be an NP-complete problem. In particular, this is almost always application dependent, so you have to carefully evaluate this each time you want to scale down your parallel algorithm. The good news here is that, starting from a work-efficient algorithm you can obtain a scaled-down work-efficient algorithm, thus preserving its efficiency.

2) To the best of my knowledge there are no strong connections with parameterized complexity, at least I am not aware of any significant published paper discussing this relation. I may be wrong of course.