The objective of this paper is twofold. Firstly, to derive time-fractional evolution equation modeling the No-Arbitrage premium of Asian option (with arithmetic and geometric averages) contingent upon an underlying asset that satisfies the fractional stochastic differential equation, in a setting when the strike price is fixed and floating. Secondly, we have computed the four versions of the put-call parities for Asian options, by solving the time-fractional Black-Scholes evolution modeling the difference of the premiums of put and call Asian options, through Fractional Reduced Differential Transform (FRDT) algorithm. We have also established the convergence and the error estimates for the FRDT Algorithm for the two independent variables