Both moves were made Friday.
Padilla was placed on the bereavement list Monday to tend to a family emergency in Nicaragua.
The Rangers said on their Web site that the release of Ponson came after off-field issues in the past few weeks.
"Based on recent events, I don't think Sidney wants to be here or deserves to be here," said General Manager Jon Daniels, who added that Ponson was put on notice after creating a disturbance at the hotel bar in St. Petersburg, Fla., during the team's last road trip.
Ponson, a journeyman, was 4-1 in nine starts (55 2/3 innings) with an ERA of 3.88, 16 walks and 25 strikeouts.
In the previous three seasons, he also pitched for the New York Yankees, St. Louis and Minnesota.