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.

The 25-year-old claimed her maiden able-bodied national title on Sunday and won team gold with England at the 2010 Commonwealth Games.
But to qualify for Rio, Brown would have to switch from her favoured compound bow to the recurve, something she will "absolutely not" do.
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"I love shooting the compound and think I still have a lot to give," she added.
"Paralympic sport is my job and I have to go out there and perform well, so that's the main thing."
Brown's stance is backed by ArcheryGB Paralympic head coach Michael Peart, a former world number two in the compound who swapped to recurve in 2001 but was never able to reach the same heights before his retirement earlier this year.
"Going to recurve would be a major change and it would take many years of training to go from the top of one [discipline] to the top of another. I tried, and in 10 years I never got there," Peart, who narrowly missed out on selection for London 2012, told BBC Sport.
"I think she has to concentrate on trying to maintain this Paralympic success and build on that stockpile of gold medals."
Brown will not be able to add to the team title she won at the 2010 Commonwealths in Delhi at Glasgow 2014 after archery was left out of the programme.
"It's pretty tough knowing that archery isn't going to be in Glasgow because I had such a great time in India and it was one of my first major able-bodied events," she told BBC Sport.
"It's a big shame, but I think it's one of those cases where you have to focus on what you can do rather than what you can't."
Brown is preparing to compete in both the able-bodied World Championships in Antalya, Turkey (29 September-6 October) and Para-archery Worlds in Bangkok, Thailand (1-7 November).
Long-term, though, her focus is on further success in 2016.
"I'm aiming for Rio and it should be amazing," she said. "It'll be my third Paralympics and hopefully third gold medal."

Lee Nicholson, 39, from Newcastle, was enlisted by Clare Humble to help dispose of Peter Hedley's body in Newburn Riverside Park in November.
Humble, 50, was jailed for life in May for murdering Mr Hedley, 49.
Nicholson, of Lydford Court, was jailed for three years and six months after admitting perverting the court of justice at Newcastle Crown Court.
Plans to cut free parking on roads around the Royal Berkshire Hospital have been opposed by a petition of more than 2,300 signatures.
[NEXT_CONCEPT]
British two-time Paralympic champion Danielle Brown has ruled out attempting to qualify for the 2016 Rio Olympics.
[NEXT_CONCEPT]
A man who helped his murderer ex-girlfriend bury her boyfriend's body in a Newcastle park has been jailed.