Parks that allow bikes:

  Ashland #
Burnside
Cascade Locks
Cave Junction
Clatskanie #
Corvallis
Enterprise
Eugene -Amazon, Churchill *
Forest Grove
Grants Pass (time restricted)
Gresham
Hood River *
Jacksonville *
Keizer
La Grande *
McMinnville
Molalla
Myrtle Point #
Oregon City
Prineville
Salem
Sandy
St Helens
St Johns
Talent #
Toledo
Vernonia
Wilsonville

*  uncertain
#  covered pegs required

There are probably more, but this is a start.
I guessed on some of the parks.
It's easy to pick up a phone and call the parks and recreation department of the city you want to ride.

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Holmes