I'm afraid I wasn't able to run your example. I believe that your fcoeff function has an error in it.
I did take a look at the Rainfall_daily.mat file. If I am right in assuming that the first column is some kind of time measure and the second is the function value, it looks like the function is extremely discontinuous.
The ODE solver used by the PDE Toolbox parabolic function has an option, MaxStep, that allows you to specify the maximum step size used during the solution. It might be possible to improve the solution you are getting by setting this to a fraction of the delta_time of your data.
Unfortunately, there is no simple way to do this without editing the PDE Toolbox parabolic function. If you edit this function (I suggest making a copy of the original first), you will see where the ode15s function is being called and where the other options to ode15s are being set. Adding the MaxStep option should be straightforward. I think this change will cause the parabolic function to execute much slower so I suggest you remove this change when you are through with this particular example.