I disagree that they're exploiting or dishonoring their murdered daughter. I take at face value that they're devastated and want to make somebody pay.
I also agree with this statement. While there have been many instances of families or estates of victims seeking restitution in some manner (as was mentioned before "treasure hunting"), this situation is one where a person in a position of trust decided to go and abuse that trust - if it were my family member who was killed, then had his or her photograph taken by a person in trust and then posted to Facebook or other social networking site, I'd be pretty f***ing upset too.
Now I'm not sure about Facebook, but I know both the mods of TMF and TT do search the various threads and if there is any posted material that violates their policies, they are removed immediately. (A good example is underage images) If Facebook also has moderators to ensure any form of compliance, (you'd think after so many issues over privacy...) the questions become this: how long was the picture on Facebook before it was noticed by the victim's family and then, based on that answer, would it have been reasonable to assume that the Facebook moderators should have noticed this and taken appropriate action? If the picture was only up for a brief amount of time then I believe it would be difficult to sue them successfully, given how many people use Facebook, how much material is posted on their servers for mods to sift through, and on top, all of the great fun Facebook has had prior re. privacy concerns. If this picture, however, was online and visible for an extended period of time (i.e. weeks) with inaction then the family may have a case as it becomes an issue of an apparent lack of due dilligence on Facebook's behalf.
I think Facebook may be somewhat liable at the end of the day, but not enough to satisfy the victim's family as it was the disgraced paramedic's choice to take the picture and upload the picture to the servers for public view. I respect the family's need for an outlet to vent their frustrations over a truly heinous act but instead of suing Facebook, and if they feel the need to sue anybody, perhaps it would be more feasible and prudent to sue the paramedic services and/or whoever pays them.
(FOR THE RECORD: ALL ABOVE STATEMENT = IMO)