My vehicle is running hot when in park?

More than likely we have a restriction in the system if it runs hot only in park.

We need to check this by doing a couple of things. First and foremost is we need to check the system for a leak by pressure testing it.

Here is a guide that will help with this:

https://www.2carpros.com/articles/radiator-pressure-test

Once we confirm there are no leaks then we need to use a laser temperature gauge to find out if we have any hot spots where the temperature drops. This would be a restriction.

https://www.2carpros.com/articles/how-to-use-an-infrared-temperature-meter

I think you are on the right path with the heater core and heater blowing cold, but the restriction is not in the parts that you have replaced so far that means we need to use the method above to find it.

Please run through this info and let us know what you find, and we can go from there.

Thanks