  The electron-electron interaction in the Landau levels of bilayer graphene is
markedly different from that of conventional semiconductors such as GaAs. We
show that in the zeroth Landau level of bilayer graphene, in the orbital which
is dominated by the non-relativistic second Landau level wave function, by
tuning the magnetic field a topological quantum phase transition from an
Abelian composite fermion to a non-Abelian parton fractional quantum Hall state
can be induced at filling factors $1/2, ~2/5$ and $3/7$. The parton states host
exotic anyons that can potentially be utilized to store and process quantum
information. Intriguingly, some of these transitions may have been observed in
a recent experiment [Huang \emph{et al.} arXiv:2105.07058].
