Some parents in Strabane said the closure of Melvin Sports Complex meant their children are missing out.
Derry City and Strabane District Council had said that it needed to ensure appropriate supervision before any changes to opening hours.
On Thursday, the council voted to carry out a report into the cost of Sunday opening.
The original proposal, brought forward by independent councillor Paul Gallagher, was to have the park opened immediately on Sundays. He said the amendment to the motion by the health and community committee was 'shameful'.
"There will be a lot of disappointed people who want access to these facilities and the council is hiding behind this umbrella of saying we need a feasibility study on the numbers who will use it and the staff costs.
"Yet we've seen this council bring forward proposals giving £3m to the airport. It's only a matter of opening up and extending the facilities."
Sinn Féin councillor Karina Carlin asked officials to compile the report before taking a final decision.
"If we're looking at Sunday opening then we have to be realistic and sensible about the matter.
"We simply can't table a motion like that and expect council staff, unions, ratepayers and everyone else to pick up the tab and the implications of that."
A report is to be carried out into the feasibility of opening a £2.2m play park in County Tyrone, on Sundays.