Your comparison to the priesthood is pretty good IMHO. Bringing up a child is a job, much more so then the distractions we commonly call work. It is a calling of the heart, the desire one was made to do, the work God intended one to rejoice in, the 9-5 rat race is a distraction from the job that will fulfill our hearts desire.
Who’s the employer? Arguably, she’s self-employed. Self-employed people have “jobs” without having “employers”, too - or rather, they are, as the term suggests, their own employer. 
What’s job-like about it? It is a necessary task, done essentially for personal/family gain, involving lots of work, as one’s full-time occupation.
When someone says “I just got a new job?” that sentence only makes sense in the context of having a person, such as an employer, who ‘gives’ jobs you can ‘get’. That necessarily excludes the self-employed.
Though more seriously - I agree that whether or not it is a “job” depends entirely on context. The real question masked by the “is it a job?” question often is something along the lines of: “is a woman who stays at home to raise a child performing a function as difficult and demanding as one who goes to work for an employer? Are these equal life choices, or should we socially favour one over the other? Is a stay-at-home parent basically a slacker, because they lack that sine qua non of modern life - a job?” and other questions of a like nature.
I agree with this.
However, just to continue the game of devil’s asshat: if a parent’s child were to be killed in a car accident, I suppose we could comfort her/him by saying, “Oh, at least you can consider yourself ‘retired’ from your parenting job. Or if you really wanted to, you could even get another job if retirement is too boring. Me, I’m stuck with the ‘job’ I got for another 18 years.”
No? Too crude? Why is that? Well, I guess the parent only wants some connotations of the word “job.” 
If you look at what L4L wrote, you will note he says it gets no sympathy from “working moms”. In other words someone who is doing the parent “job” in addition to a income “job”.
Which brings up a pet peeve of mine. Why is the assumption that only women are doing the work of raising children still the norm?
Couldn’t the same sort of “Job’s Comforter” comfort a person whose employer has gone bankrupt, tossing them out on their ear without a prospect? “Hey, early retirement!”
Seriously, though, what a stay-at-home parent does with him or herself after the kids are grown could be a bit of an issue, if they had kids young.
There is no doubt being a single parent (man or woman) and having to raise a kid and work, too, is extra-tough.
Being a “working mom” though (or for that matter, “working dad”) - that all depends; if they are in a couple, and how much money they are making, and what arrangements they have for child-care - when my wife had her job, we earned enough (specifically, from her salary) for a live-in nanny, and still made a profit. I doubt either of us would be getting much sympathy from anyone …
The word “job” was not handed down by God meaning what it means. Through most of the world, and most of human history, the majority of people have not participated in formal work-for-wages labor. Is running your own farm a “job?” Because that is mostly what humans have been doing, and you can bet that raising the farmhands…er, I mean kids…preserving food for the winter, sewing and maintaining the clothes, etc. was every bit as much of a “job” as planting the wheat and feeding the cows.
Even in the modern industrialized world, there exists ambiguity about the meaning of “job.” Is an unpaid internship with a professional company and a 9-5 work schedule a “job?” Is going to grad school a “job?” Does it make a difference if you get a stipend vs. getting a wage? How about full-time volunteer opportunities like the Peace Corps?
Anyway, I think parenting is essential work that does contribute to society, and it should be recognized as that. Things have changed a lot, and it’s no longer “the woman takes care of the house so that the man is capable of devoting his attention to work,” but parenting and housework still contribute to the overall ability prosperity of society (not just the individual involved, like a hobby) and should be afforded some respect.
Of course that can apply to stay at home parents as well. If you have a house keeper and send your kids to an exclusive pre-school, how tough is your “job”? On the other hand, there are stay at home parents dealing with children with issues who work harder and longer than most people collecting a salary. There is no set level of effort required to parent. Some parents are lucky, some are lazy, and some just suck.
Yup, absolutely. One cannot tell from a simple label like “stay at home mom/dad”, “working mom/dad”, etc. how hard (or easy) one has it.
I love my work, but it is still work and still a job. Those who imply that stay-at-home moms don’t work have no idea of the effort this entails - we have evidence for this.
I think “hobby” is not giving credit to the decision making process for some who stay at home. Not getting into which option is really better, don’t you think that some stay at home because they feel it maximizes their child’s development, not because it is amusing?
Why? Saying a stay at home mom works hard is not implying that one working outside the home isn’t working hard?
Absolutely it’s a job. No question about it.
Filing back in Pandora’s Box under ‘things I wish I’d never asked’.