An explosion that damaged the office in Crawford Square on Monday night was caught on CCTV.
Police said they were given an "inadequate" warning before the device went off.
Finbarr O'Doherty, secretary of the area's residents association, said the office should be moved to a more secure area.
"The probation office should be moved to another location, not right stuck in the middle of Crawford Square.
"In my opinion they have every right to be there but it was attacked before with stones and paint bombs and I think it should be on government land, in a more secure area.
"Its right beside two houses with young families and there's no guarantee that this won't happen again."
Laverne Mason, a mother of two young children, lives next door to the office and said she and her family were terrified.
"I just started shouting fire to the kids and once I got them downstairs it was just a matter of getting out because I thought the car was going to blow up.
"I was just screaming because I just didn't know what was happening, it was just totally out of my control."
Ms Mason said she now feels unsafe in the house that has been her family's home for more than 50 years.
"We need to take immediate action in regards to getting the probation office removed.
"We have had years of problems with them, they are aware of this and I'm just very disappointed today that they haven't even directly contacted us to see if we are ok.
"Someone, myself or my children, God forbid, could have been killed last night and they need to be removed from this street."
In a statement, the Probation Board for Northern Ireland said: "We work in the heart of communities to deliver a public service and have worked in Crawford Square for more than 30 years.
"We understand the concerns of local residents following this incident on Monday night and we are mindful that police are continuing to work to understand the motives for this attack.
"We will continue to work with the police to ensure our offices are safe for those who use them and live nearby."
Neighbours of the probation office in Londonderry have called for it to be moved, after a bomb attack.