You should be straight with whoever you are talking to in HR and ask them what the opportunities are in that location. In some places the turnover is so high you might actually have a crack at part time supervisor pretty quickly. But the places with high turnover are usually that way for a reason.
One thing to know is that a part time supervisor gets health benefits right away. Part time hourly have to work a year. That's the current setup - it might change in the new contract in August, but you don't know. Right now the management have to pay for benefits and hourly get them provided as part of the contract - but here again that might change with the new contract. The starting part time hourly pay is pretty low right now. It will be higher with the new contract, but maybe not as high as the $15 being asked for. The supervisor pay rate is not affected by the contract, so you might not end up making a whole lot more as a part time supervisor than you would as an hourly, but that's all in the future.
But if you are considering the health benefits and you thought you could be a supervisor inside a year, it might be worth it.
A part time supervisor supervises a group of part time hourly employees. The job depends on what operation you're on. Preload is going to start early in the morning. Other operations are different depending on the location. You could be working a day sort, a twilight or a night sort and have different hours. Part time supervisors normally work 27.5 hours a week.
I would say that your age and background is an advantage. I have seen quite a few part time supes in their early 20s who have a hard time. They don't always have good experience dealing with people in stressful situations and they crack and just start treating people disrespectfully. Being able to stay cool, listen to people and treat them with respect is just better for everyone.
Going into full time management is also going to depend on your location. There are two pretty common paths - one is to go from part time hourly to part time supervisor to full time supervisor. People who do that are usually in inside operations and you end up as a hub or dispatch supervisor, but it is possible to go from part time operations supervisor into other areas, like security. The highest-paid starting full time supervisor job is on-road supervisor. To do that you have to pass driving school and do some driving usually - even if it's only as a seasonal driver. UPS also likes to promote full-time drivers to on-road supervisor, so that is another route.
It's going to depend on the location, and that's where you want to talk to HR and even better to someone who already works there. Some places might have a shortage of management and that would be the fastest way up. Or they might be looking for full time drivers and you'd have a chance to bid a driver job quickly, then go into management from there (and that's not a bad option if you're needing income and benefits). I've seen it both ways - part time supervisors stuck for 10 years waiting for a chance, and part time hourly waiting 10 years for drivers to retire to get a shot to drive. But then I saw a part time preloader go from off the street to part time supervisor in under a year and full time within a year of that.