Does FEATool solve multiphysics problems in parallel?

Our in-house FEA codes in MATLAB do not scale well to high-fidelity mesh resolutions. In the past we have used some in-house large-scale codes in C++ that have worked well, but to have an FEA solver that can do large-scale parallel computing while interfacing with MATLAB would be very nice. I've looked on the FeniCS website and they mention high-performance parallel computing through python.

The FEATool Multiphysics distribution by default comes with the "mumps" linear solver which features shared memory OpenMP parallelization, unfortunately there are sometimes severe compatibilities for Matlab versions >=2020 so is therefore not used by default on such systems. So in theory you can get significant speedup with mumps but it may depend on your system configuration how well it works.

The FEniCS backend with the PetSc solver framework (direct, iterative, multigrid etc.) as you point out can offer significantly more in terms of performance, and allows for distributed computations via MPI parallelization. The drawback is that simulations have to be "piped" to Python and FEniCS and back, however the additional overhead should be mostly negligible compared to the potential performance gains.

So a general recommendation would be to start and see if and how much mumps helps, and then use FEniCS to potentially get further performance gains.