Define the properties of a grammar that is the fastest to parse
Your question is unclear. When you write "parse", do you actually mean
to produce a parse structure (one, or all possible ones). How do you define
it ? For example, some grammars have a parse tree that differs from the
derivation tree. Or did you mean that the language must be fasted to
recognize?  Regular grammars give you languages that are recognized
in linear time. What qualifies as a grammar? Actually, if you are just
interested in recognition, you are looking for properties of languages
rather than properties of grammar (though they can be related).

You can certainly build hierarchies of grammars according to
complexity criteria if you clarify the above.

For example in the case of Context-Free grammar, the is nearly a well
defined concept of parsing (up to specifying whether you want one
parse tree or all possible parse trees).

The complexity is at worst $O(n^3)$ for parsing and $O(n^{2.373})$ (due to valiant, with a looser bound at the time) for
recognition (wich is only asymptotic, and not the fastest in any
realistic situation). There are CF grammar that will parse in linear
time (notably deterministic grammars), others  will parse in $O(n^2)$
time, and others are believed to require more (though the exact
complexity bound is not known), even for simple recognition.

I do not really know if these complexity hierarchies have been
identified precisely.

The usual hierarchies studied for context-free grammars are often more
related to the applicability of techniques that can be used to build
parsers for these grammars.

As the question is stated, I guess there is one language that is faster than any other to recognize. That is the Babel language invented (nearly) by Jorge Luis Borges and containing all possible string (Borges was unfortunately limited to a finite size due to paper shortage, which would invalidate what I am saying). With all strings of any size, recognition is easy. You start in an accepting state and terminate in 0 steps.
The empty language is recognized as fast. I believe these are the only ones recognized in 0 steps (for a given alphabet). This establishes that, if the alphabet is fixed, the class of languages recognizable in 0 time is closed under complementation.

This is of course in jest, but you can see from it that one can define languagesn and corresponding grammars, that can be recognized in constant time, by putting constraints only on a finite prefix of strings. They are not much more interesting. The problem is in the statement of the question.

I guess the fasted interesting class of CF grammars is the class parsable in linear-time. Deterministic CF grammars are clearly member of that class, so that is a sufficient property. However many non-deterministic languages, having no deterministic grammars, can also be parsed in linear time.