  We consider a setup consisting of two coupled sheets of bilayer graphene in
the regime of strong spin-orbit interaction, where electrostatic confinement is
used to create an array of effective quantum wires. We show that for suitable
interwire couplings the system supports a topological insulator phase
exhibiting Kramers partners of gapless helical edge states, while the
additional presence of a small in-plane magnetic field and weak
proximity-induced superconductivity leads to the emergence of zero-energy
Majorana corner states at all four corners of a rectangular sample, indicating
the transition to a second-order topological superconducting phase. The
presence of strong electron-electron interactions is shown to promote the above
phases to their exotic fractional counterparts. In particular, we find that the
system supports a fractional topological insulator phase exhibiting
fractionally charged gapless edge states and a fractional second-order
topological superconducting phase exhibiting zero-energy $\mathbb{Z}_{2m}$
parafermion corner states, where $m$ is an odd integer determined by the
position of the chemical potential.
