  We prove linear convergence for a new family of modified Dirichlet--Neumann
methods applied to quasilinear parabolic equations, as well as the convergence
of the Robin--Robin method. Such nonoverlapping domain decomposition methods
are commonly employed for the parallelization of partial differential equation
solvers. Convergence has been extensively studied for elliptic equations, but
in the case of parabolic equations there are hardly any convergence results
that are not relying on strong regularity assumptions. Hence, we construct a
new framework for analyzing domain decomposition methods applied to quasilinear
parabolic problems, based on fractional time derivatives and time-dependent
Steklov--Poincar\'e operators. The convergence analysis is conducted without
assuming restrictive regularity assumptions on the solutions or the numerical
iterates. We also prove that these continuous convergence results extend to the
discrete case obtained when combining domain decompositions with space-time
finite elements.
