You definitely do not want to shoot regular 12 bore loads in a badly pitted barrel. I have a set of their earlier version that simply fits in the chamber for a 1880's era 10 bore. The barrels are fine, but the old style 11/4 ounce 10 bore loads are really hard to find. I can then shoot modern 12 loads out it for clays.
The new version which you mention are about six-inches long, and if your gunsmith says that will take the pressure safely past the damaged area, then you should be fine.
Depending on the model of your Parker, the only other option is to have the barrels sleeved. Not sure who does that on fine guns in the US these days, but that essentially removes the damaged barrels forward of the chamber. The last time I had a gun done was about fifteen years ago on a fine London sidelock. Back then it was over two grand and only made sense on a fine gun (as expensive). Depending on the grade of your Parker, that might make economic sense.