There is a Recall on Jags (S206-R166), because (and that discusses) the Electric Park Brake (EPB), as it can become "unintentionally applied", your issue may be related to the same problem. Also, an actual Jag Tech would know how to do a reset on your EPB system to check the EPB failure (there is a reset that involves a disconnect of the battery, and there is a Technical Bulletin that advises how to check the EPB issues which discusses the "How To" of that reset.) I almost forgot until I saw a note I had saved myself in my saved photos. There is also a National Highway Traffic Safety Administration Recall [see safercar dot gov] related to the 'Power Train' (Jaguar Recall R525; NHTSA No. 05V053000) that discusses that the vehicle may show as being in Park, while not actually being in Park, due to an issue with the "Electronic Transmission Park Prawl Guide Plate", although this seems like it is opposite of what you state as your current vehicle issue, I recall reading a BUNCH of owner posts on various sites, very similar to your own, which were resolved by that owner fixing this recall's issue. Having said that, this recall is old enough that this "fix" should have been fixed already on your vehicle if it were a problem (but, you never know with age/wear if it just kicked in). You can contact Jaguar Directly via Support (800-452-4827) & using your vehicle VIN No to check their records (or the NTHSA 1-888-327-4236), they can tell you if a Jaguar Dealership any where performed any regular maintenance or recall maintenance in their/Jaguar record. Even if a recall correction was performed through other than a Jaguar dealership, Jaguar would still have on record that recall fix was performed in that specific vehicle as the recall parts would have had to have been requested from them/Jaguar, and would still be tied back to your vehicle's VIN No.
While this isn't a "How To" answer, I do hope it helps you.