Nature-inspired algorithms are becoming popular due to their ability to solve complex optimization and engineering problems. Grey Wolf algorithm is one of the recent nature-inspired algorithms that have obtained inspiration from leadership hierarchy and hunting mechanisms of grey wolves. In this paper, four formulations of multi-objective grey wolf algorithm have been developed by using combination of weighted objectives, use of secondary storage for managing possible solutions and use of Genetic Algorithm (GA). These formulations are applied for jobs scheduling on parallel machines while taking care of bi-criteria namely maximum tardiness and weighted flow time. It has been empirically verified that GA based multi-objective Grey Wolf algorithms leads to better results as compared to their counterparts. Also the use of combination of secondary storage and GA further improves the resulting schedule. The proposed algorithms are compared to some of the existing algorithms, and empirically found to be better. The results are validated by numerical illustrations and statistical tests.