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I have a '98 with a 100k miles and I went through all the troubleshooting described above but the button never released. Pulling fuse 21 did nothing to release button tension. My local speed shop found the problem in 15 minutes. There is apparently a cable that comes off the back of the ignition switch that releases the push button also. He said it was a manual cable and not an electrical cable. This may be part of the anti theft circuit. He omitted this cable so now the car can be shifted out of park without a key but he left the solenoid circuit in place so you still must have your foot on the brake to shift.
This started out an as intermittent problem over the last 6 months. Usually after messing with it the button eventually depressed. Then it may go several months with no problems at all. It finally quit working all together but fortunately I was at home.
Seems like an alternate resolution. Temperatures/climates... If you have an A4, then you can simply disconnect the BTSI under the "shifter" boot. In fact, will not "throw a code". Easy shift from P to D & fun... Less security, VATS will be it. That and a .45 So far, have added Column (CLB) and BTSI (unplugged) so as not to be stranded. So far, so good. Months ago. Not worried. Next will be another attempt to "trick" the car into believing it still has F45 suspension. LG GT2s going on Monday. Will circumvent the F45 with similar resistance. Can't wait! Document your process. Be safe.