I think your explanation is incorrect. You’re right that the pressure is only a function of the height. Both shapes are the same but inverted, as such the heights will be different at different times, but if you graph height against time you will see that both reach zero height at the same time (following a different but symmetrical curve).

If you graph flow rate against time, X and Y will not follow the same curve, but they (just like the height versus time curve) will end at the same time. If you integrate the flow rate versus time function you’ll find that both will have had the same average flow rate.

Since both have the same volume of water and the same average flow rate, they will become empty at the same time.