Subaru won't start, cranks slowly, even when jumped

Hi, I'm unable to get my 1990 Subaru Legacy (2.2L 4-cyl, 298,000mi) to start. Here's the progression of events, hopefully someone has some ideas.
My daughter borrowed my car and accidentally left the parking lights on. She ran the battery down, but apparently not enough to keep her from starting the car and driving it to my house. I turned the lights off when she got here, before she shut the engine off. About 30 minutes later, I tried to start the car, and got a bunch of click click clicks in response. This has happened before on this car/battery, when I had a dead battery, so I got a jump from a fella in a big, new pickup truck. The subie started no problem. I drove it around to charge the battery (this usually works), then shut the engine off, and immediately tried a restart. All I got were clicks. Got another jump from another big pick-up, no problem. Car started fine. Drove home. Turned off, turned on, click. The battery appeared to be dead. When I would try to start the subie without jumping it, the lights would dim, and the car would 'reset'. When I say reset, I mean the seatbelt alarm would beep and all the warning lights would come one, as if I had I just turned the key to ON for the first time.

Here's what I've tried so far:
1. Cleaned the battery terminals (they were corroded). Tried jumping. Same veeeeerrrrrrryyyy sloooooow cranking. Wouldn't start.
2. Charged battery overnight using a home battery charger (that converts 120AC to 12VDC). The engine cranked over a few times and almost started when I did this, but the battery quickly died.
3. Tried the combo of a charged battery and jumping the subie with my running V8 mustang (Connected negative jumper of subie to engine grounding strap). Cranked over a few times at almost normal speed, but didn't catch. Engine slowed quickly. Couldn't get it to turn over more than once or twice when I turned the key to START, and it wasn't turning over anywhere near fast enough to actually start.
4. Tried a new 800/640A battery and jumping at the same time. No dice. Cranked fast enough to start for a couple seconds, but didn't. Cranking slowed within about 3 seconds to very slow. New battery is dead.

My plugs and cables are both only a couple years old, and the plugs have been gapped and cleaned within the year.

Thoughts? Could this be my starter? Do I need to battery cables? I have a multimeter, so I could check resistance. I'm stumped.
Josh