This does not make sense. The parking brake disengages as soon as you start driving. Essentially acting as "hill start assist". I do this all the time. Although I don't know what happens when you pull parking brake button while diving...I had the exact same problem as the OP. Got the "overheated" warning and then got the "Service Required" warning. I took it to Ford for service and they cleared all the error codes and couldn't find anything wrong.
I just got home and noticed the tech wrote on the invoice "highly suspect vehicle being driven with parking brake on causing overheating issue" Give me a break. I've been driving for 35 years, probably longer than the technician has been alive, and I know how to drive, and I would know (within 3 seconds) if I was attempting to drive with the brake on.
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