Age of Legolas, is father Thranduil and grandfather Oropher is quite unclear. Tolkien Gateway i found this:

Unfinished Tales - Celeborn and Galadriel - Appendix B said:

Oropher was one of the Iathrim who spent his early life in Doriath.[1] ``` and ```Thranduil was one of the Iathrim who spent his early life in Doriath.[1]

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But in the source they give only Oropher is mentioned:

Unfinished Tales - Celeborn and Galadriel - Appendix B said:

Oropher had come among them with only a handful of Sindar, and they were soon merged with the Silvan Elves, adopting their language and taking names of Silvan form and style. This they did deliberately; for they (and other similar adventurers forgotten in the legends or only briefly named) came from Doriath after its ruin and had no desire to leave Middle-earth,

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It doesn't even say that Oropher was born before the ruin of Doriath F.A. 507 like Tolkien Gatway assumes. It only says the folk that was led by Oropher came from Doriath. It might as well be true that Oropher's unmentioned father or grandfather was from Doriath and he himself was born during the first 1000 years of Second Age.
Thranduils date of Birth is even more unclear. In the quote above it seems that Oropher arrived in Mirkwood 1000 S.A. without Thranduil (but ofc he could just not be mentioned). I believe that many people understand this quote like Thranduil was one of the elves who arrived in Mirkwood:

RotK - Appendix B said:

but before the building of the Barad-dûr many of the Sindar passed eastward, and some established realms in the forests far away, where their people were mostly Silvan Elves. Thranduil, king in the north of Greenwood the Great, was one of these.

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But to me it seems that it refers to his anchestry only and doesn't mean he himself was among the elves who arrived in Mirkwod. Why? He wasn't king of the woodland realm in Second Age. His father Oropher was king until he died in the Battle of the Last Alliance.
Legolas Date of Birth is even more unclear. He is not mentioned at all before the Events of the Lord of the Rings. The only hint I can find is this:

Unfinished Tales - Celeborn and Galadriel - Appendix B said:

Oropher was slain in the first assault upon Mordor, rushing forward at the head of his most doughty warriors before Gil-galad had given the signal for the advance. Thranduil his son survived, but when the war ended and Sauron was slain (as it seemed) he led back home barely a third of the army that had marched to war.

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Legolas is not mentioned in the events of the Battle of the Last Alliance. One could read it like that he was not born yet or not yet old enough to take part in the battle. But again he could also just not be mentioned.
So in summary:
Oropher could be born any time before Second Age 800 (to be old enough to lead the goup of elves that arrived in Mirkwood). But I agree it seems more likely he was born in Doriath before First Age 507.
Thranduil could be born any time before Second Age 3350 (to be old enough to take part in the Battle of the Las Alliance). It seems likely to me that he was not born before Second Age 1000 (as he is not mentioned to arrive with Oropher in Mirwood).
Legolas could be born any time before Third Age 2900 (to be old enough to be send to Elrond's Council). It seems likely to me that he was born in the early Third Age (as he's not mentioned in the Battle of the Last Alliance).