If I understand your set-up correctly, there is no direct implication between the two conditions.   But Sasha Premet can give a more definitive answer, having been a pioneer in the study of finite $W$-algebras.  (As Yves comments, it would help to provide more precise background.)   I assume you are looking at finite $W$-algebras in the spirit of Losev's 2010 ICM survey <a href="http://front.math.ucdavis.edu/1003.5811">here</a>
and related literature.  Here it's conventional to leave aside the extreme case $e=0$.   Many nilpotents are of "standard Levi type": principal (also called regular) in a Levi subalgebra of some parabolic subalgebra of the given simple Lie algebra, but outside type $A_n$ there are other nilpotents as well.   

The most standard examples of *even* gradings of $\mathfrak{g}$ arise directly from the adjoint action of any three-dimensional simple subalgebra containing $e$ and correspond to Dynkin diagrams for nilpotent orbits having only even labels.  Already in type $A_n$, one sees non--even orbits even though all nilpotents are of standard Levi type.    
 
On the other hand, in other simple types there are examples of even nilpotents which are not of standard Levi type, such as the subregular nilpotents in type $D_4$.   

The more general "good" gradings were classified by Elashvili-Kac and developed further by Brundan-Goodwin <a href="http://front.math.ucdavis.edu/0510.5205">here</a>.  But the connection with parabolics seems to be narrower here than in the notion of standard Levi type.   (By the way, it's tricky to sort out in the Dynkin-Kostant classification those nilpotent orbits which are of standard Levi type, as seen already in the subregular case.)  

P.S. Although you are interested in characteristic 0, essentially everything remains the same in good prime characteristic.