Chomsky Hierarchy and P vs NP
Regular languages
Regular languages are in $\mathbf{P}$ because a deterministic finite automaton is a restricted deterministic Turing machine that runs in linear time.
Context-free languages
In fact, any context-free language is in $\mathbf{P}$: Valiant showed in the 1970s that context-free grammars in Chomsky normal form can be parsed in time $O(n^3)$ [1]. $\mathbf{P}$ strictly includes the set of context-free languages, since $\{a^nb^nc^nd^n\mid n\geq 0\}$ is not context-free but is clearly in $\mathbf{P}$.
Context-sensitive languages
Context-sensitive languages can be parsed in nondeterministic linear space [2]. We don't know the exact relationship between this class and $\mathbf{NP}$ but, since a linear space Turing machine can use exponential time, probably there are context-sensitive languages that aren't in $\mathbf{NP}$. However, showing this would be a major advance in complexity theory as it would imply that $\mathbf{NP}\neq\mathbf{PSPACE}$.
The problem of &quot;Here's a context-sensitive grammar $G$ and a string $w$: is $w\in L(G)$&quot; is $\mathbf{PSPACE}$-complete [2], so certainly every problem in $\mathbf{NP}$ can be reduced to the parsing problem for context-sensitive gramars. However, that's not the same thing as saying that every language in $\mathbf{NP}$ is defined by a context-sensitive grammar, since the $\mathbf{PSPACE}$-completeness result allows arbitrary polynomial-time computation in the reduction and also allows the grammar to depend on the instance. Hopefully, somebody can post a comment or answer clarifying this.
Unrestricted grammars
Unrestricted grammars define all recursively-enumerable languages. This includes all of $\mathbf{NP}$ and includes undecidable languages such as the halting problem, which definitely aren't in $\mathbf{NP}$.

References
[1] L. G. Valiant, General context-free recognition in less than cubic time, Journal of Computer and System Sciences 10(2):308–315, 1974.
[2] Context-sensitive language, Wikipedia.