The parity transformation encodes spin models in the low-energy subspace of a larger Hilbert space with constraints on a planar lattice. Applying the quantum approximate optimization algorithm (QAOA), the constraints can either be enforced explicitly, by energy penalties, or implicitly, by restricting the dynamics to the low-energy subspace via the driver Hamiltonian. While the explicit approach allows for parallelization with a system-size-independent circuit depth, we show that the implicit approach exhibits better QAOA performance. We propose a generalization of the two approaches in order to improve the QAOA performance while keeping the circuit parallelizable. Furthermore, we introduce a modular parallelization method that partitions the circuit into clusters of subcircuits with fixed maximal circuit depth, relevant for scaling up to large system sizes.