Why do you think it was important for slave owners to keep slaves ignorant about their birthdays and parentage? Douglass opens his story in Narrative of the Life of Frederick Douglass, an American Slave by telling us that he is troubled by not knowing when he was born. Why is this fact so important to him?

When Douglass says that slaves know as little about the circumstances or dates of their births as horses do, that one small comparison of enslaved human beings to a farm animal suggests a lot about why slave owners kept their slaves' birthdays and parentage from them.
The slave owners wanted to dehumanize the slaves as much as possible. As Douglass himself points out, slave owners saw slaves inquiring about their birthday as having "restless spirit"s—these are people who want to be individuals and refuse to be seen as expendable chattel.
The reasoning is much the same with keeping slave parentage secret. People feel a sense of identity with their families. Even today, people love knowing where their ancestors came from and their cultural heritage. Slaves were denied this basic comfort. Once again, slave owners did not want the slaves to develop personhood. They wanted to keep them in submission to their owners. Even something as basic as a parent-child relationship stood in the way of their profits.

Douglass notes:

I have no accurate knowledge of my age, never having seen any authentic record containing it. By far the larger part of the slaves know as little of their ages as horses know theirs, and it is the wish of most masters within my knowledge to keep their slaves thus ignorant.

He goes on to say that his master "deemed all such inquiries [about dates of birth] on the part of a slave improper and impertinent, and evidence of a restless spirit." To know one's date of birth means that it is a date worth acknowledging. A birthday is a celebration of life, an acknowledgement of value and a place of worth.
It is likely that slave owners simply didn't value the lives or the births of slave children enough to make a note of the date. The fact that they owned slaves capable of the necessary work was enough. What practical or economic purpose could knowing an exact date of birth serve these owners? Since they had almost nothing to personally gain from keeping track of these records and viewed their slaves as merely acquired property, the date of birth was insignificant.
If masters did know a date of birth, they likely didn't share this knowledge with their slaves because doing so would create a feeling of significance regarding this date. Most slave owners tried to belittle and demean their slaves in order to maintain the sense of order they desired, so they didn't want to create a sense of individuality.
Parentage was kept quiet for other reasons. Douglass, for example, believes that his father was one of his white owners. This situation happened frequently and was not a source of joy or celebration for slave owners. While they might have gained one more slave to own, they certainly did not claim these offspring as their children. It was also common for masters and owners to disperse biological families across different plantations. After all, families make sacrifices for each other and form bonds that could overcome powerful obstacles. By breaking apart families, slave owners weakened the collective power of the slaves on their plantations.

Douglass opens his book saying he was unhappy, even as a child, not knowing the exact day of his birth. He doesn't say exactly why this was troubling, perhaps assuming it was so understandable that he did not have to explain it. He notes that the white children's awareness of their birthdays was a "privilege" he never had. The word privilege is a hint at what he is thinking.
Not telling the slaves the day on which they were born is a way of dehumanizing them, and Douglass intuits this from an early age. It is another way of separating them from whites and emphasizing that they are less valuable. They are treated more like farm animals or livestock, which don't need to have a specific birthday attached to them. They are not treated as individuals whose lives and histories are inherently worth celebrating and remembering. They are only valued for the profit they bring their owners. This dehumanization helps keep the slaves demoralized and reminded that they are "inferior."
This is a subtle form of dehumanization, but one Douglass is acutely aware of. Today, we might call it a micro-aggression.