I don't think it's disrespectful, as long as it's in her own room, although it does make ones's eyes roll.  I had a similar issue with younger SD referring to the time NPD BM was with DH repeatedly as "the golden years" on Facebook, although they were nothing of the sort of course, except in her mind.  She was age 4 when her parents split, now age 22 she is still trying to wrangle them back into each other's company.   She recently invited DH on a weekend trip with her and NPD BM to visit other SD where she lives, up north.  Who knows, maybe NPD put her up to it?  

apparently SD wants BF and BM back together to play Happy Family. And wants you out of the picture. You must Disengage from SD. You don’t take, do, buy, for crazy SD.  She is never going to like or respect you.  So don’t play her game 

Personality disordered people often don't really let go of their exes, and vacillate between love and hate for them. A breakup is a common time to circle back to an ex and again see him as "the one that got away".  These daughters are caught up in their mothers' craziness, in both directions - the love and the hate.

Allow her the picture, but do get her therapy..

I would not be bothered by the picture as long as it stays in SD's room.
It is something she wants to hold on to - your husband doesn't.

TBH I differ form a lot of the others. I would be bothered by it. Though a lot of my frustration comes from MIL trying to tell me that I was "requried to hang pictures of Psycho and DH together in our home. So the kids won't feel their family is divided." Which REALLY turned me off from it. THen to top it off, cleaning out storage and finding pictures mixed in with things definitley wasn't fun either. And DH thankfully got rid of all of them. BUt between that and Psycho's obvious mistreatment of the girls. I would be completley bothered by that picture not existing, but being displayed in my home.

It's not necesarilly anger related, but with all she's put all four of us (DH, myself, and the girls) through, and continues to put us through. I honestly woudn't be happy with a picture of her in my home in general, with or without DH in it. Maybe if the girls want one later they could have one of just Psycho so long as it stays in their rooms. But it wouldn't be welcome to me. Particularly one like that.

However a lot of this is probably situational, it's hard to know how I'd feel, or how the girls would feel, if the situation was different, or if she was a good, sane, person who actually cared about the wellbeing of the girls, or who at the very least did the bare minimum for their wellbeing. Initially I thought everyone could be sane, and I feel in that siuation I might feel different about this topic.

Did BM put her up to it? Sounds like it to me. I have a pretty good feeling that the picture doesn’t necessarily mean much to SD but was a manipulation ploy by BM.  “Here is a picture of me and your dad for your room at dads house!”, I would say that picture was given knowing the reaction it would cause- and I also feel that SD probably doesn’t really see anything much to it.  By her age kids are usually fully adjusted once parents have been apart for a while, and the only ones I encounter who aren’t are ones with personality disordered bio parents who want to keep the kid psychologically fixated on their parents relationship in order to manipulate the other parent further.

My DD is the same age, and has friends whose parents are divorced as well and literally none of them could care less about their parents relationship. All they care about is that their parents don’t fight, and that it doesn’t impact their day to day lives.  Kids at this age are pretty self centred for the most part and could not be bothered by “romance between their parents” or “what was”- they just don’t care about that sort of thing. Kids don’t even want to think about that sort of thing and usually find it gross. 

I would say that mom is trying to have skid fixated and act as her little manipulator by proxy. She is going through an emotionally hard time because of her second divorce and she is reminiscing and grasping at straws. 

What I would do is give zero reaction to it- if you do it just goes back to BM and she will be filled with glee that she can still have an impact and think that she is capable of causing trouble and then will up the antics. You could also just slip the photo into a drawer if looking at it bugs you- I would say it doesn’t matter at all if it was just a normal photo but because it is a kissing one I honestly believe she was put up to it. Or you could get DH to say that it is an inappropriate photo and to keep it in a drawer.

For Christmas I would send mom a photo of you and DH under the mistletoe. hahah.

I drew the line at a photo that suddenly appeared of SD and BM back when they lived in my house. I asked SD to replace it with a more current one, I would be happy to take the photo of them and pay to have it printed and framed. I just didn't want "remember when" of the good old days photos in my home. It became obvious that SD purposefully put the photo up for that purpose when she pitched a fit, refused the offer of a new photo and took ALL her photos down. *shrug* 

while I fully support having photos of the other parent in their room, I refuse to have photos used as weaponized memories to make me uncomfortable in my own home. 

(just to clarify, it was a photo of them snuggling in the house)

I think that if games are going on with BM and the skids are still confused about having two different homes then pictures from the good old days are not a good idea.  Pictures with a parent are ok and reinforce the idea that the relationship with each is respected and need to be kept in the skid's room.  

OP, its your home and its up to the adults in that home to set the tone of it.  Exes don't get to do that and don't ever let them.