You’re right.
From the Score layout, go to Layout Options, ensure that the Full score layout is selected in the right side of the dialog, then turn Bar Rests back on (from Players > Bar Rests and Multi-bar rests). Apply and Close.
Grab the rest that appears in the piano staff at bar 6 and turn off its Ends Voice property.
Grab the rest that appears in the piano staff at bar 15 and turn off its Ends Voice property, too.
Then go back to Layout Options and turn off Bar Rests again.
With any luck the score now looks like this:
and the piano part looks like this:
If you’re unsure about Starts Voice and Ends Voice, it’s probably the result of using Edit > Remove Rests without understanding how that works: Remove Rests sets those properties on the notes and rests either side of the passage whose Rests it’s removing. In early versions of Dorico those properties existed; Edit > Remove Rests was added as a timesaver but uses the earlier properties to achieve the same result as doing it by hand.
Alternatively it may be that you’ve imported in some funky MusicXML, in which case you might consider going into Preferences > MusicXML Import and unticking the Rests > Rest visibility option. This won’t fix this file but will prevent this from happening the next time you import MusicXML.