Jenny and Ashley Bonsall said they were angry about the "disgusting" thefts from their daughter's grave
Councillors and police say they are trying to stop thefts and vandalism at children's graves in Herefordshire.
Nearly 1,700 people signed a petition calling for CCTV to be installed after the incidents at Hereford Crematorium.
Ashley Bonsall, who began the petition, said he was "enraged" by thefts from his baby daughter's grave.
Herefordshire Council said low lighting at the site meant CCTV may not be the answer but it was "working to come up with an effective solution".
For more on this story and other Herefordshire and Worcestershire news
Two thefts of "sentimental ornaments" from graves were reported to officers in February, while earlier this month four male youths were issued warning letters following reports of anti-social behaviour at the crematorium in Westfaling Street, West Mercia Police said.
'Stoop so low'
Mr Bonsall said a fairy ornament which had sentimental value for his family was taken from the grave of his daughter, Krystal Marie, who died aged 30 weeks.
"For someone to take it is just disgusting - I can't believe some people would stoop so low as to do this," he said.
Herefordshire Council said discussions over the provision of CCTV at the site were under way.
"The operation of CCTV in the area is difficult due to the low light and as to whether it would be effective or not - so we are working to come up with an effective solution to help prevent this type of unacceptable behaviour again," a spokesman said.
Meanwhile, police appealed for anyone with information about the incidents to contact the force and said officers from Hereford Safer Neighbourhood Team would be carrying out "reassurance patrols".
"Thefts from cemeteries are completely thoughtless and cause a huge amount of upset and distress to the families whose loved ones' graves have been affected," a spokeswoman said.
