The proposals mean charges would be placed on 625 spaces in the area and restrictions placed on a further 741.
Reading Borough Council has spent four years designing the scheme "to relieve parking pressures at the hospital" and to make parking easier for residents.
The petition, set up by hospital staff, describes the scheme as "problematic".
The online petition opposes the introduction of pay and display meters for Addington Road, Erleigh Road and other roads around the hospital in Reading.
Clare Goulbourn-Lay, a Royal Berkshire Hospital midwife, set up the petition to express her concerns about the potential consequences of the scheme.
"By making changes to this road, you're not just making it difficult for the staff, you're making it difficult for the patients and pushing the problem further out," she said.
"With the university so close as well, there simply isn't enough parking space as it is. To add another restriction is just silly."
She has called for the scheme to be cancelled or redesigned to "help make spending time with loved ones easier" and to limit the "stress and anxiety" of staff and patients.
John Sharpe, who lives on Erleigh Road, has called the plans "divisive" and said he is concerned the restrictions will make parking pressures worse.
"I'm scared it will become impossible to live in the area," he added.
Reading Borough Council said "up to five years of public consultation had taken place on the principle of prioritising parking for residents in the area" and that further public input on the plans are being sought.
Residents have until 13 June to submit their views to the council.
Plans to cut free parking on roads around the Royal Berkshire Hospital have been opposed by a petition of more than 2,300 signatures.