I think each job, working parent or SAHP, has its own set of challenges and difficulties. I'm a SAHM. It's more emotionally exhausting than many other jobs are. On the surface, it doesn't seem hard. It's honestly not all that much work for me to keep the house neat and the children alive. But the emotional exhaustion has me completely wrung out by their bedtime.
The first time my husband stayed home with The Child while I was gone for the day, I came home to find him half asleep on the couch, feeding the baby, and he said, "I don't know how you do this all day." It was gratifying. But I could easily say the same to him. I don't know how he can be gone from the house for ten hours a day and then come home and be a father with all the energy that requires.
I'm no more a chef, housekeeper, stylist, and daycare worker than my husband is an IT tech just because he can fix some of his own computer glitches while he's at work. And my job isn't easy just because I don't do much physical or mental work, any more than his job is easy just because he doesn't have to deal with his coworkers' tantrums, bumps, bruises, and naptimes.
They're each jobs, and each carries its own challenges and difficulties. I prefer to be a SAHM. My husband and I both planned for that when we got married. Sometimes I hate it, but overall I much prefer to be at home with the kids. The SAHP v. working parent argument isn't terribly productive because it's comparing apples to oranges. I can't compare the difficulties of the two any better than I can compare the difficulties between construction work and accounting.
ETA: I'm not saying they can't ever be or shouldn't be discussed or compared. But I have never seen this discussion not devolve into a heated, us v. them snipe-fest. If it can be done anywhere, I'm sure it's eHell, but it's a potentially contentious topic, especially when approached from the wrong angles.