The proposals to build the car park on Green Flag-accredited Boston Manor Park have already been given support in principle by Hounslow Council's executive committee.
But councillors have said that while they support the plans for the park in Brentford, west London, they do not wish to lose any green space.
The car park could be built in a section of the park that is already concreted beneath an elevated section of the M4.
Friends of Boston Manor Park liaison officer Linda Massey said the organisation was frustrated with a lack of consultation.
"Local residents are particularly upset about losing the view and we often run festivals in that section," she explained.
"Having a car park there is not conducive to a nice walk around the park."
Hounslow Council has now agreed to hear from the Friends of Boston Manor Park and Linda Massey will present arguments to local councillors at an area committee meeting on Thursday (5 March).
A report from the council states that parking at GSK House, though less than the company wishes, already comfortably exceeds the maximum standards set out for new development in both Hounslow's unitary development plan and the London Plan.
However, it is concerned about finding funding elsewhere during the recession.