  The current proposals for producing non-Abelian anyons and Majorana
particles, which are neither fermions nor bosons, are primarily based on the
realization of topological superconductivity in two dimensions. We show
theoretically that the unique Landau level structure of bilayer graphene
provides a new possible avenue for achieving such exotic particles.
Specifically, we demonstrate the feasibility of a "parton" fractional quantum
Hall (FQH) state, which supports non-Abelian particles without the usual
topological superconductivity. Furthermore, we advance this state as the
fundamental explanation of the puzzling $1/2$ FQH effect observed in bilayer
graphene [Kim {\em et al.}, Nano Lett. {\bf 15}, 7445 (2015)], and predict that
it will also occur in trilayer graphene. We indicate experimental signatures
that differentiate the parton state from other candidate non-Abelian FQH states
and predict that a transverse electric field can induce a topological quantum
phase transition between two distinct non-Abelian FQH states.
