NP-hard in what parameter?
A problem is always claimed to be NP-hard, period.


Indeed, a problem's definition already contains a specification of its parameters. (See the entries in Richard Karp's seminal collection of NP-complete problems for several examples.) Usually, there is no need to make explicit reference to the parameters per se, as they are "automatically scaled" by the input length; for your two parameters $m$ and $n$, for instance, the input could simply be encoded as $0^m1^n$. Hence, for abstract problem instances, it usually suffices to refer strictly to the input length, which is indeed how time complexity is defined (i.e., as the number of computation steps as a function of the input length).

Studying how different requirements on the parameters impact the problem's complexity is the object of study of parameterized complexity theory. For example, the set of SAT formulas with $k$ many variables, $k$ being a fixed parameter (i.e., constant), is decidable in polynomial time (in the formula's length); in parameterized complexity terms, this means SAT is fixed-parameter tractable. It is only as the number of variables is also allowed to scale (in the input length) that SAT becomes NP-hard.