What is actually the task here? Parsing is clear and it looks like one has to sum all parsed timestamps, but what does the additional comment about the shortest time mean?

It means that your solution is scored on how fast it runs.

Frequently we end up with solutions that use tricks that would not be applicable to the n=1 problem in TFA ("parse 1 timestamp") when we try to get the fastest solutions for "find the sum of all these timestamps."