The regulations at the park have changed over this last summer to be consistent with WA State mountain pass chain requirements.
Here's the deal simply stated: You need to have chains in your car when you come into the park - even if you don't use them and the posted restriction is "snow tires advised".
The reason for this is that the road conditions rapidly change during a snow event. So even if you drove to Paradise in the sunshine, when you leave there may be 6 inches of snow on the road.
And if you have a 4x4, remember that actual tire restriction is "Tire Chains Required - 4 Wheel Drive with approved traction tires OK - 4 Wheel drives must carry chains".
This carry chain requirement for 4 wheel drive vehicles is likely to be enforced at chain-up stations.
Here's the tire restriction FAQ off the NPS.GOV website.
Tire Chain Requirement - Frequently Asked Questions