The circuit model of a quantum computer consists of sequences of gate operations between quantum bits (qubits), drawn from a universal family of discrete operations(1). The ability to execute parallel entangling quantum gates offers efficiency gains in numerous quantum circuits(2-4), as well as for entire algorithms-such as Shor's factoring algorithm(5)-and quantum simulations(6,7). In circuits such as full adders and multiple-control Toffoli gates, parallelism can provide an exponential improvement in overall execution time through the divide-and-conquer technique(8). More importantly, quantum gate parallelism is essential for fault-tolerant error correction of qubits that suffer from idle errors(9,10). However, the implementation of parallel quantum gates is complicated by potential crosstalk, especially between qubits that are fully connected by a common-mode bus, such as in Coulomb-coupled trapped atomic ions(11,12) or cavity-coupled superconducting transmons(13). Here we present experimental results for parallel two-qubit entangling gates in an array of fully connected trapped Yb-171(+) ion qubits. We perform a one-bit full-addition operation on a quantum computer using a depth-four quantum circuit(4,14,15), where circuit depth denotes the number of runtime steps required. Our method exploits the power of highly connected qubit systems using classical control techniques and will help to speed up quantum circuits and achieve fault tolerance in trapped-ion quantum computers.