"Does your child's father have warrants?" the Sunday morning tweet read. "Is he carrying around any drugs? Has he been committing any crimes? Want to give him a Father's Day he'll never forget? Call TPD and we'll help your family make a memory that will last a lifetime."
The tweet was deleted soon after it was published.
In a statement Monday, the department said a member of its social media team posted the tweet which "was meant to be humorous and light-hearted, but was hurtful to some ... We would like to sincerely apologize for this mistake.
"This will be a learning experience and goes to show that words, even when intended to be humorous, can be hurtful to others."
The department statement concluded by thanking people who reached out to say how the tweet made them feel.