We derive sharp estimates on the modulus of continuity for solutions of a large class of quasilinear isotropic parabolic equations on smooth metric measure spaces, with Dirichlet or Neumann boundary conditions if the boundary is non-empty. We also derive optimal lower bounds for the first Dirichlet eigenvalue of a class of homogeneous quasilinear operators, which include non-variational operators. The main feature is that this class of quasilinear operators has corresponding one-dimensional operators, which allow sharp comparisons with solutions of one-dimensional equations. © 2024 International Press, Inc.. All rights reserved.