The odds of both parents and their baby being born on the same date is 1 in 133,000. This phenomenon is quite rare, but it does happen. It is possible for a baby to be born on the same birthday as one or both of its parents, but it is not a common occurrence. The probability of this happening depends on the number of children the parents have and the distribution of birthdays in the general population. 

For example, if the parents have one child and the chance of any given birthday occurring is equally likely (1/365 for a non-leap year), then the probability of the child being born on the same day as one of the parents is 1/365. If the parents have two children, the probability increases to 2/365, and so on.

It is also worth noting that the probability of two people having the same birthday is higher in a group of people than in a single family. 

This is known as the birthday paradox. In a group of 23 people, there is about a 50% chance that two people will have the same birthday. This probability increases as the size of the group increases. In some cases, the parents may have shared the same nursery at the hospital where they were born, adding another special layer of coincidence. The baby could also be named after their parent’s shared birthday, making it a unique and special moment for the family.