Parc Howard in Llanelli costs Carmarthenshire council more than Â£100,000 to maintain each year.
But Parc Howard Association said the building, which houses a museum, and parkland needed more investment.
Councillor Meryl Gravell said one option would be for somebody else to run park.
Parc Howard was built in 1885 and given to the town of Llanelli in 1912 by Sir Stafford and Lady Howard.
The grade II-listed mansion houses a collection of Llanelli pottery, artwork and town history.
The association said it was worried for the future of the Italianate country house and 24 acres of parkland, adding they were starting to deteriorate.
Ken Rees, chairman of Parc Howard Association, told S4C's Newyddion 9 programme: "If they [the council] aren't ready to spend money to maintain the place it's going to become an eyesore."
He added: "This park is the jewel in the crown of Llanelli."
Ms Gravell, Carmarthenshire council's executive board member with responsibility for regeneration and leisure, admitted the future of the park was "uncertain".
"At present the only other option is perhaps finding someone to come and take the park over," she added.
A 19th Century mansion and parkland in Carmarthenshire are at risk of becoming eyesores because of council cuts, according to those who help run them.