The use of a triad of seasons may have something to do with the mild winters in Southern Greece. Specifically that they don't have a "proper winter" as those inhabiting northern climes would understand it. As oerkelens points out, some cultures divide the years into two seasons, rainy and dry, which is purely based on climate.
The triad represents the three key aspects of the cycle of seasons in an agricultural context, namely Θαλλώ ("blossom"), Αὐξώ ("grow") and Καρπώ ("fruit").
Triads, particularly involving goddesses, were quite prevalent in Greek Mythology, so that may also have been a factor.
Note that these words can be variously translated, but the translation I choose gives you the gist. I've linked to the lexical entries on Perseus so you can take a look at the range of meanings for each word. It's worth mentioning that Robert Graves in the index to The Greek Myths translates Καρπώ as "withering", which certainly has more of a wintery flavor and would seem to acknowledge Persephone's time in the underworld.