It is challenging to build scalable quantum processors capable of both parallel control and local operation of qubits. As a promising platform to overcome this challenge, optical lattices offer exceptional parallelism. However, they are limited in achieving precise local operations due to relatively narrow lattice spacings. Here, we introduce a quantum processor that combines orbit-qubit encoding with internal states as auxiliary degrees of freedom, enabling spatially selective operations alongside parallel control. With this processor, we generate one- and two-dimensional cluster states using minimal layers of controlled-Z gates. We experimentally detect the multipartite entanglement of a two-dimensional cluster state involving 123 orbit qubits through direct stabilizer measurements, verifying the full bipartite non-separability. Furthermore, we demonstrate measurement-based quantum computation by implementing single- and two-qubit logical gates, highlighting the flexibility of orbit-qubit operations. Our results establish orbit-qubit optical lattices as a scalable architecture for quantum processing, opening new pathways for quantum computation applications.