The Bit Error Rate (BER) comparison of Adaptive Parallel Interference Cancellation (APIC) algorithms in asynchronous Multicarrier Direct Sequence Code Division Multiple Access (MC-DS-CDMA) system is introduced. The APIC is used to eliminate the Multiple Access Interference (MAI) in MC-DS-CDMA system. The Adaptive filtering algorithms considered are, Normalized Least Mean Squares (NLMS), Variable Step Size Normalized Least Mean Squares (VSS-NLMS), and Generalized Normalized Gradient Descent (GNGD) algorithms. These algorithms are used in coincidence with APIC system for updating cancellation weights. The performance comparison between these algorithms is performed via extensive computer simulations. Simulation results demonstrate that the VSSNLMS algorithm outperforms all other algorithms, the NLMS is modular, while the worst performance is obtained by GNGD.