If a body with an accurate clock is moving away from A which is stationary, then the time in B would be slower than that in A. Since relative to B A would have an equal velocity time in A would be slower than that in B. This results in a contradiction. This is the argument given in this website. Can someone say how this argument is wrong from a deeper study of SR.?
Time dilation is real and has been experimentally tested (http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/airtim.html).
The example you provide is unintuitive but not contradictory. Consider A observes B moving at some velocity $v$. Now B also observes A moving at some velocity $v$. Both observations are correct without contradiction. A contradiction only occurs if you assume there is some "absolute" notion of velocity. Such an assumption would contradict many of our laws in physics since the laws remain invariant if you shift the velocity by some constant. Similarly, a contradiction in time dilation only occurs if you assume there is some "absolute" notion of time.
This results in a contradiction
A observes B's clock to run slow, B observes A's clock to run slow. It seems contradictory but it isn't.
It would only be a contradiction if simultaneity were absolute but it isn't, simultaneity is relative.
For A to observe the rate of a clock stationary in B's frame, A must use two spatially separated and synchronized clocks stationary in A's frame.
Similarly, for B to observe the rate of a clock stationary in A's frame, B must use two spatially separated and synchronized clocks stationary in B's frame.
Why? To record the start and end time on B's clock, A must have a clock co-located with B's clock at the start and at the end. But B's clock is moving in A's frame; B's clock location has changed between the start and end times. Thus, A requires two spatially separated clocks and these must be synchronized according to A for the measurement to be valid.
And there's the resolution of the contradiction: A observes B's spatially separated clocks to be unsynchronized and vica versa.
This is so well known and so well understood and this question has been asked so many times here and elsewhere, I suspect your question will be closed in short order.