Ben Atkinson launched his petition on Friday to ask police to "reconsider how they approach their search for offenders".
"There have been numerous posts by the Canterbury Police searching for offenders in the community, where it acts as a platform for members of the community to bully and humiliate the offenders," Atkinson said.
"We live in a society where we struggle immensely with mental health and cyber-bullying. This is 'illegal' but somehow permitted on the subsidiary pages of the NZ Police."
Atkinson said potential solutions could include police changing its Facebook pages to groups to be able to disallow comments by the public, and by having a diverse and encompassing list of keywords to be banned by the page.
The petition, which Atkinson says will be delivered to the NZPCA and a District Court judge, has already attracted more than 2200 supporters.
Canterbury Police posted on its Facebook page on Friday asking people not to abuse or harass wanted individuals.
New Zealand Police deputy chief executive of public affairs Karen Jones said they have talked with Atkinson to "better understand his concerns and how they can be addressed".
Social media is an important tool for police, with more than 750,000 followers on its Facebook pages.
Jones says it assists police find individuals, seek information, provide reassurance, get prevention messages out, and inform its communities about what police are doing.
"It also helps us engage directly with a wide range of people. We must balance these positive benefits against the potential for inappropriate, abusive or hurtful comments to be made," Jones said.
"This is a situation faced by anyone who engages in public-facing social media channels, not just police.
"This particular post attracted a large amount of 'trolling' from members of the public which was very disappointing and the post was subsequently taken down."
Comments cannot be turned off on business Facebook pages. This can only be done on group pages.
Police have a moderation filter running on all of its Facebook pages that automatically hides comments that contain profanities and bad language. They update the list of words regularly to ensure all offensive language is hidden.
"However, it is still possible to troll people online without using profanities," Jones said.
"We do our best to hide comments that do not comply with our guidelines for appropriate behaviour, but as this is a manual process, it can take some time when there are hundreds of comments to moderate."
Police say they will continue to use social media but in response to the recent concerns, Jones said: "We will look at the options for how we do 'wanted' postings to ensure we best meet our operational objectives, while at the same time ensuring individuals are not subject to unwarranted comment."
