The NBA is the National Basketball Association. So the judicial variety of court is a place where a judge presides and people are tried; and the NBA variety of court is a basketball court (more generally, sports courts). Variety means sort .
ps - this is a bit of a rant, but this sort of writing really 'gets my goat' [irritates me]. If you're only interested in the original question, ignore this!
The sentence is trying to be clever and managing only to be badly written and confusing. The parenthesis '(both the judicial and the NBA variety)' doesn't work, because although it's possible to use 'court' (the judicial kind) to mean 'everyone concerned with and/or all the processes to do with the court', the same doesn't apply to the sports kind of court. If the writer wanted to use this reference to two different sorts of court, he should have used the preposition in; there are two problems with this:
1. Psychologists don't work in all the other things in the list (which are groups of people and/or enterprises).As he's already listed 'athletics teams', he could make this item more general and get rid of the parenthesis altogether:
2. They don't work in basketball courts
Psychologists also work for sports teams, engineering firms, consulting firms, and the courts.- shorter and clearer, and the only sacrifice is a quasi-literary device that doesn't work.