Your first sentence sounds fine and natural to me.
Started in 1987, the festival exhibits more than 550 varieties of mangoes and provides a rare opportunity to taste them all for free.
We are standing in the present ("exhibits" and "provides") looking back into the past ("started"). In the past it was started, now it exhibits. It's clear that the festival itself started in 1987, and that the present activities are "exhibiting" and "providing".
Your second sentence sounds a little odd.
Starting in 1987, the festival exhibits more than 550 varieties of mangoes and provides a rare opportunity to taste them all for free.
We are in the past ("starting") progressing towards the future (or the present, or the more recent past). Then suddenly we're in the present ("exhibits" and "provides") and (grammatically speaking) we don't know how we got here! It sounds like the "exhibiting" and "providing" started in 1987, not the festival.
A third way is:
Starting in 1987, the festival (has) exhibited more than 550 varieties of mangoes and (has) provided a rare opportunity to taste them all for free.
Now everything happens in the past, however, I don't think this is what you want. It would appear this festival is still being run, so you want to use the present tense. Of course there are other tenses you could use as well as some that wouldn't work, and whichever tenses you decide to use should be consistent with each other. Fix your grammatical feet in once base tense (most likely the present), and go from there.