This paper introduces an improved iterative soft decision subcarrier parallel interference cancellation (SDSub-PIC) technique for the carrier interferometry/multicarrier code division multiple access (CI/MC-CDMA) system that significantly reduces the multiple access interference (MAI) for the desired user. Carrier interferometry (CI) codes are used to minimize the cross-correlation between different users. In this paper, the interference cancellation is done by taking the soft decision estimates of transmitted data bit at the subcarrier level. MAI estimates become more reliable with multistage iterative structure of the receiver, which ensures improved bit error rate (BER) performance. Simulation results show that SDSub-PIC provides average BER of 1e-04 at 10 dB SNR which is about 1.5 dB off from single user bound in an AWGN channel at 50% overloading. In slow frequency selective Rayleigh fading channel, SDSub-PIC ensures BER of 4e-05 at 25 dB SNR with four-fold diversity at 100% overloading. We have observed that the new scheme performs considerably better than Block-PIC [14] and Sub-PIC [15] proposed earlier for CI/MC-CDMA system. Complexity of SDSub-PIC is O(N) while for conventional PIC and Block-PIC, it is O(2N-1) and 2O(N) respectively.