The para- prefix indicates that a complexity class is parameterized by some other measure of complexity. Specifically, a language L is in the parameterized class para-C, if there is an alphabet A and a computable function pi(k) -> A*, such that every (Q,k) is in para-L if and only if (Q,pi(k)) in L.
The prototypical example (as well the violation of the naming convention) is para-P, which is almost always known as FPT, which is equal to DTIME(f(k)n^c) for some constant c.
Space-parameterized examples include para-L and para-NL, which are equal to DSPACE(f(k)+log(n)) and NDSPACE(f(k)+log(n)), respectively.
Compare with the slicewise complexity classes X-, such as X-.
Discussed in:J. Flum and M. Grohe. Describing parameterized complexity classes. Information and Computation.andElberfeld M., Stockhusen C., Tantau T. (2012) On the Space Complexity of Parameterized Problems.
para- version of L. Equivalent to DSPACE(f(k)+log(n)) for some computable function f. Compare with slicewise parameterized logspace, XL.
Parameterized vertex cover (is there a vertex cover of size at most k) is complete for para-L. (Elberfeld et al, 2012)
para- version of NL. Equivalent to NDSPACE(f(k)+log(n)) for some computable function f. Compare with slicewise parameterized nondeterministic logspace, XNL.
It seems open whether there are natural complete problems for para-NL. However, the related class para-NL[f log] has many natural complete problems.
para-P is a less common name for FPT, but in line with other para- classes naming conventions. Its slicewise counterpart is still called XP.