Direct sum theorems state that the cost of solving $k$ instances of a problem is at least $\Omega(k)$ times
the cost of solving a single instance. We prove the first such results in the randomised parity
decision tree model. We show that a direct sum theorem holds whenever (1) the lower bound for
parity decision trees is proved using the discrepancy method ; or (2) the lower bound is proved
relative to a product distribution.