The first 3 main games follow the same story, so to fully understand the story and all of its references, I would recommend playing the first and second game (in order) before playing the third. I would not say it is necessary, because it is an action-oriented game, and each part plays fine on its own.
Both games were very well received, so it is certainly no punishment to play through them, if you can look past the age. I think they aged quite well, but it has been some time since I have touched them.
For the sake of completeness: God of War (2018) features the same protagonist and loosely continues the story. Whereas the first three 'main' games are direct sequels in terms of a continuing story, the fourth main entry (named simply 'God of War') is set years later, in a different setting.