The three Fates are named Clotho, Lachesis, and Atropos; and each one deals with a different part of life. Since there are three, they inevitably came to be associated with past, present, and future; but I don't think that's entirely accurate. The descriptions I've seen generally refer to Clotho as the one who spins the thread for each mortal, Lachesis as the one who determines the length of a person's life, and Atropos as the bringer of death by means of her thread-cutting shears. Perhaps it's more accurate to say that they represent the beginning, middle, and end of life; rather than past, present, and future specifically.
The idea of three spinning women who determine destinies appears in some other mythologies as well. Norse mythology has the Norns, who serve a similar purpose. Actually, sources refer to an entire race of Norns, female deities who have the power to influence the lives of humans. It's possible that the specific idea of three spinning Norns was borrowed from the Greeks, but it's hard to tell. Latvian mythology has Laima, Kārta and Dēkla, another three sisters who determine the fates of mankind, but I don't know that they were ever associated with spinning. And the three wyrd sisters from Shakespeare's Macbeth were obviously a continuation of the same theme, and they really DO specifically reference the past, present, and future.