At most birthday parties we attend, the birthday kid doesn't open presents from their own parents. That is usually done at home with just immediate family.
In our case, we don't have any family that lives nearby, and I absolutely hate planning parties. My boys each got a party for their first birthday, then another one when they turned 5. Maybe we will done one when they turn 10, I don't know.
My kids do get plenty of presents from the grandparents, so we generally do only small gifts for them. My older son turns 8 in October, and I recently found an awesome Lego clearance sale at Target so I got him a set for $35. I think that is the most I ever spent on a birthday present for him. Younger son turns 6 next week. I found a Paw Patrol game on the clearance rack for $7 that I know he will love. DH doesn't think that is enough, so he is going to buy another gift for him. But in general, we don't buy a lot of gifts for the kiddos, and they get so much from others that they really don't care.
Just keep in mind, that you are setting the precedent now. If you give small gifts at 2, then continue to give small gifts each year, then that is just what they consider "normal." If you give bigger gifts at 2, 3, 4, etc. then try to scale back when they are 8, 9, 10 - then they are much more likely to notice and wonder why mom and dad are being "mean" or not giving them what they want.