Jumps are taught in a very specific order, that helps skaters build the skills and muscle memory they need as they become more complex. You'll see skaters practice the jumps in this order as well: Waltz jump, salchow, toe loop, loop, flip, lutz, axel (or axel lutz). Skaters will usually start with singles and then move up to the double, then the triple jumps when practicing. Even the top skaters generally start with a simple waltz jump.
But jump technique actually starts way ahead even of a waltz jump, with a simple bunny hop and the various half-jumps. So here's the tutorial:
The half jumps:
Bunny hop, is a simple forward take off and landing, off the toe of the gliding foot onto the toe of the free foot, which then pushes back to the original gliding foot. It's the first named jump taught. Leg action translates directly to axel entry, so it's important to get it right. Bunny hop is the scariest jump in skating, because, yikes two toe picks, both forward. You can also do a "double" bunny hop, essentially skipping on the ice. Bunny hop teaches skaters basic upper body movement, proper leg action in a jump, and how to rock to the toe pick for jump take offs.
Half flip, half toe walley, ballet jump, scissor or mazurka, half lutz are all 1/2 rotation versions of the full jumps (ballet jump has same entry as toe loop). The entries translate very directly to the full rotation versions, so proper entry technique on these is extremely important. These jumps are often used in footwork sequences (you're not allowed to do a full rotation jump in footwork).
Full rotation jumps
Of the full rotation jumps, only the lutz really goes 360 degrees. The others have slight pre-rotations on the take off (I'll get arguments about this on the flip). These pre-rotations are not "cheated," they are part of the jump technique. You can cheat an entry, but some jumps actually incorporate a small forward edge. You know you're good at jump ID when you can spot the difference.
But this also helps fellow travelers like parents understand why jumps are taught in a specific order. Generally, jumps are taught bunny hop, waltz jump, half jumps, salchow, toe loop, loop, flip, lutz, axel. No one teaches walleys or mazurkas anymore. Doubles are added in the same order. Each jump has slightly greater rotation than the one preceeding it. Lutzes, not axels, are considered the most difficult of all the jumps because the entry edge is the opposite rotational direction to the air rotation (known as a "counter" jump). The challenging part of the axel is the extra half-rotation, requiring both good timing and serious upper body strength.
There are a lot of jumps that haven't made it into the cannon so to speak. An inside axel (launches off the inside edge instead of the outside edge). Walleys and toe walleys (a counter jump off an inside edge). There's a wonderful jump variation off a flip-- 1 1/2 flip, which lands forwards. Split lutz-- full rotation with an aerial split in the middle. Many of these jumps are taught in the ISI curriculum, which is one of the reasons I really like ISI for high level skaters.
There is no set amount of time for learning these jumps. There is no "a skater must be doing xx jump by age yy or they can't compete." Master each jump and then move on. Don't move on before you've mastered it.
Do parents need to know the jumps?
Not really. But it's fun to test yourself.