I experienced this exact problem in my '86 300E, but only during cold (below freezing) weather. Starting in neutral always solved the problem. I highly doubt you have any kind of starter/solenoid problem if this is the case with you. My guess is that there is a sensor going bad that tells the computer the car is in Park, and the lockout is in effect....and from my experience the sensor is temperature-sensitive. I'm not a mechanic in any way, shape or form...just using logic here.

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1996 SL320 97K miles
1996 C220 130k miles
1992 500SEL 170k miles <---sold
1986 300E 216k miles <---sold