Multi-valued logic potentially increases the efficiency of arithmetic circuits and digital signal processing. Quaternary logic can be suggested as a solution to the problem of power consumption and connection complexity in binary digital systems. The possibility of having several threshold voltage levels using the carbon nanotube field-effect transistor results in the widespread use of this technology in the design of multi-valued circuits. Here, a quaternary full adder cell is proposed. The main goal of the proposed circuit is to reduce the critical delay path in the quaternary full adder circuit using a parallel design. The proposed circuit is compared with four quaternary full adders based on simulation results using HSPICE in 32 nm technology. The experimental results show the higher performance of the proposed quaternary full adder cell than the state-of-the-art designs.