The "trails" for horses (and bicycles) on Rattlesnake Mountain are actually all old logging roads, as the footpath carved by various local trail organizations is for hikers only. That's fine, because the footpath is very steep. Between the logging roads and the powerline maintenance roads, there are still miles to ride that offer their own views. Note that ongoing or recent logging sometimes changes what roads are accessible.
The trailhead starts in Snoqualmie Point Park by the map near the restrooms; horses are permitted on the path until you reach the logging road intersection in a few hundred feet, then you'll turn left on the gravel road while the hikers go straight ahead. (A right turn takes you shortly back to a blue gate and the paved road you came in on, a possible alternative if the parking lot gate is closed.)
View Rattlesnake Mountain in a larger map
To Snoqualmie Point Park: take I-90 east to Exit 27. At the end of the off-ramp, go right and follow the road to the end. Go through the gate on the right to enter the trailhead parking lot. Loop around the parking lot, the trailer parking is a wider spot on the opposite side from the restrooms. If the gate is closed and you have to park on the road, there is a gravel pathway starting next to the gate alongside the road up to the parking lot.
Length: many miles of old logging roads and powerline maintenance roads
Surface: gravel, occasionally quite rough gravel; not recommended for unshod horses
Share with: bicycles, hikers, dogs
Parking for: 3-4 trailers; more
along the entry road if you're bold
Snoqualmie Point Park belongs to the town of Snoqualmie:
Since this is a relatively new trail, not many maps are online, but the Green Trails map for Rattlesnake Mountain (purchasable at local hiking supply stores) should show the logging roads with fair accuracy. Some of the lesser-maintained trails on that map no longer exist, though.
King County Parks has added a downloadable "brochure" (including a map) for this park to their "Backcountry Trails Map Brochure Series."