Hi,

Unfortunately you are looking at an internal transmission failure here. While it is possible to have for instance low fluid level that would cause the same condition in Drive, it would not explain your vehicle slipping in Park. If this were the case, you would have to have a mechanical failure of the Parking lock as well.

Your most likely scenario is you have either a burnt out Low Reverse Brake or a physical fracture of either the LR annulus gear or OD planetary carrier.... both of which are controlled by the Low Reverse Brake.

The reason this is the most likely situation is that Low Reverse Brake is used both for first gear, and for holding the transmission when in Park. If the Low Reverse Brake is unable to apply (via hydraulic failure, solenoid failure, or clutch disc failure), you will have a bad slip in first gear only and you would be unable to lock the transmission when in Park.
If the Low Reverse Brake were working, but there was a physical failure of the LR annulus or OD planetary carrier, the transmission would attempt to do it's job locking the Low Reverse Brake, but doing so would not actually activate anything due to a failure of one or both of the two gears it controls (again annulus or planetary).

If you are not experiencing any Service Engine Soon light or flashing Neutral light, it is most likely going to be a gear failure. If you have only the flashing Neutral light it could go either way (gear or LRB control failure), and if the Service Engine Soon light is on, it could go either way but might lean a bit toward the LRB control side. Obviously having the codes checked from both the Service Engine Soon light and the transmission (requires a special scan tool, generic readers won't work) would be the first step in diagnosis. I would expect to see ratio failure codes present from first gear slipping, but would want to see if there were any others that might lead us in one direction (LR solenoid failure, for example).
If nothing unusual was found there, the next step would be a fluid pressure test on all circuits, specifically LR though, and followed by a transmission removal and tear down for inspection.