Should I play God of War 1 & 2 before starting GoW3

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.

It is a matter of opinion, but I would say No. I played GoW III before going back to GoW I myself and still haven't gotten to GoW II. While you do miss a bit of the backstory if you play GoW III without playing I & II, there is not a lot of story there to start with (to be clear, I think what story is there is good and well done, but it is still a thin story). I think you will miss very little of the enjoyment of playing GoW III by skipping I & II.

Also, as Ben helpfully pointed out, other sources including Wikipedia provide detailed summaries of the backstory from GoW I and GoW II.