A perk to bringing your child under the age of 3 to the Walt Disney parks is that they do not need a park ticket to enter any of the theme parks. If you have booked multi-day tickets, you will not need to purchase a ticket for your child if they turn 3 during your trip, as long as they were two on the first day you visited the parks. (Cast Members will tell you that nobody ages Disney!) Additionally, children under age 3 will not need a Fastpass to enter an attraction with their party. Lastly, children under three are welcome to share from your plate at Disney restaurants, and will not be required to be on a dining plan.
Disney does not require any proof of age for children under age 3. Occasionally, a Cast Member will ask the age of the child as they are passing through the main entrance. If you have a 2 year old that is tall for their age, feel free to make a copy of their birth certificate and bring it with you, however no one I have ever encountered has been asked to provide proof of age.