  This work constructs the first-ever sixth-order exponential Runge--Kutta
(ExpRK) methods for the time integration of stiff parabolic PDEs. First, we
leverage the exponential B-series theory to restate the stiff order conditions
for ExpRK methods of arbitrary order based on an essential set of trees only.
Then, we explicitly provide the 36 order conditions required for sixth-order
methods and present convergence results. In addition, we are able to solve the
36 stiff order conditions in both their weak and strong forms, resulting in two
families of sixth-order parallel stages ExpRK schemes. Interestingly, while
these new schemes require a high number of stages, they can be implemented
efficiently similar to the cost of a 6-stage method. Numerical experiments are
given to confirm the accuracy and efficiency of the new schemes.
