Hi Sarah! Sounds like a great plan. Hiking up from Parson’s will be tough — no way of sugar-coating that. It’s a steep fire road with no switchbacks. But personally I think it will be easier than going down (where you’re constantly about to slip and fall on your butt!).
I think PL to LH will be a long haul, so start early. LH to BJ will be easier, and BJ to Avalon shouldn’t be that bad even though the mileage is long, mainly because you start high and end at sea level.
I wouldn’t personally worry about your stuff at Parsons. Setup your camp and don’t leave anything like your wallet, phone or camera behind and you should be fine. The lockers at Parsons are not “re-lockable” as I recall.
Keep us posted and let us know how it goes!