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.
Notable deaths of 2014 [PHOTOS]
Julius Randle breaks leg during NBA debut