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Parallel operations are important for both near-term quantum computers and larger-scale fault-tolerant machines because they reduce execution time and qubit idling. This study proposes and implements a pairwise-parallel gate scheme on a trapped-ion quantum computer. The gates are driven simultaneously on different sets of orthogonal motional modes of a trapped-ion chain. This work demonstrates the utility of this scheme by creating a Greenberger-Horne-Zeilinger (GHZ) state in one step using parallel gates with one overlapping qubit. It also shows its advantage for circuits by implementing a digital quantum simulation of the dynamics of an interacting spin system, the transverse-field Ising model. This method effectively extends the available gate depth by up to two times with no overhead when no overlapping qubit is involved, apart from additional initial cooling. This scheme can be easily applied to different trapped-ion qubits and gate schemes, broadly enhancing the capabilities of trapped-ion quantum computers.