From GMSV
At the west entrance of the NP, there are OR62 reassurance shields in both directions as the county boundary is marked -
link. The pavement changes roughly here too (suggesting a maintenance border).
At the south entrance there's no OR62 shields. Though there are reassurance markers for the Volcanic Legacy Scenic Byway. One might expect begin/end signs if Oregon didn't view the road as continuing through the national park.
Where the main road to the lake turns off, there's
this sign which says "Highway 62 Jct".
NPS's maps of the NP mostly don't have OR62 marked in the park, but
this one does mark it.
I'm going to say that, while there's a break in state maintenance, the gap in OR62 is merely administrative, with both Oregon and the NPS viewing the road through the park as being part of Highway 62.