My caravan shifts into park under hard acceration

refers to shift switch mechanism under dash or down on the tranny

have a pressure test done on the transmission to check for internal problems. You may need a band /clutch pack adjustment as it may be slipping a bit before engagement. May a good service will fix the faults as dark tan oil is a sign of troubles inside.

look under truck on drivers side around area under drivers seat you will see transmission linkage mounted on side of tranny this connection known to get corroded or damaged causing problems, from hard to shift from & to park or no shift at all

nuetral safety switch has issues or the button is not depressed at the brake lever--if u go under and look it can be seen and felt--press it in to start car

Maybe something wrong with neutral switch? Usually it is on or close to transaxle shift lever, follow the shift cable to shift lever, the switch has multiple wire connector.
It may be possible to start vehicle by jumping the load side of starter relay, key on. There are safety issues trying this, the vehicle will crank in any gear. Make sure tranny is out of gear, parking brake is set, block the wheels, whatever it takes to be safe, safety is your responsibility.
If the problem isn't that neutral switch, then I don't know? Any applicable trouble codes?

Is it over filled?

You cold try cleaning the valve body

I guess you have a modulator,is that
leaking fluid,pull of vacuum hose & check

Quite often that will happen when vehicle parked with wheels turned, and/or vehicle on a slope, so try to avoid that in future.If nothing else, when parked on a slope, stop, keep foot on brake pedal, apply parking brake, put transmission in neutral, then take foot off brake pedal and wait for parking brake to stop vehicle rolling, then shift it into park. When leaving, put foot on brake pedal, shift to drive or reverse whatever direction your going, then release parking brake. Sounds like a lot to remember but you won't have to pull hard on shifter as before, and it could save a lot of money on transmission repairs down the road. Always try to park with wheels straight as well.
Now, with that said, if you parked with wheels turned, try hard to turn steering wheel in same direction wheels are turning, just slightly, as you try to turn key. Sometimes a quick yank in that direction will do it. Also make sure shifter is all the way in park before you try it. If your on a slope try rocking vehicle slightly up slope as you push/pull shift lever in to park and try to turn key at same time. If no luck, try pushing brake pedal real hard and try to turn key. Let me know how it goes.

Things haven't changed much...look on top of the transmission in the engine compartment and look at the shifting linkage. Look for anything that could be interfering with the shifting. One linkage is the cable to the Shift Lever and Rod that goes down into the transmission. Remove the cable adjustment screw.
Move the shift lever rod towards the front of the Van into the Park Indent Position. Place the Driver's Position (Dash) Shift Lever into Park and then reconnect the Cable to the Shift Lever and tighten down the cable adjustment screw to 70 in-lbs.
Check operation's
Let me know if this helped, or if you have any additional information, questions of problems, Feel free to contact me at FixYa.com!

you will need to replace both pieces of the ignition switch assembly,i suggest you let the dealer do this job

The transmission is probably shot. the only other thing could be dirty linkage if you mean "hard" as in hard to move the shift lever. The first thing I would do is get a tranny fluid and filter change.