Does anyone know if you can have dogs on the hiking trails in Chautauqua Park in Boulder?

Short answer - Yes -
From the City of Boulder Website:
Chautauqua Area Dog Regulations
All dogs must be leashed in the Trailhead Leash Area. Beyond the trailhead, dog control requirements vary from trail to trail, so carefully watch for dog regulation signs at trailheads and trail intersections. On most OSMP trails in this area, dogs must be on a hand-held leash at all times unless they meet the voice and sight control standard and display a City of Boulder Voice and Sight tag. Dog excrement removal is required by law. A dog station is available to aid in the collection of dog excrement.