From the Burger King in Moab, drive SW along Hwy 191 7.4 miles to a junction with the La Sal Loop Road. Drive up the La Sal Loop road for 12 miles, then take a dirt road on the right towards Geyser Pass. Drive on this for 5.55 miles and take a right onto a smaller dirt road. Follow this another 1.65 miles to the end. In winter, the road is plowed to a plowed parking lot just before the junction with the Gold Basin Road.
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This is the easiest route up Peale in winter, and will be described as a winter route (since that's when I was there). It can also be done in summer. You will need to read the topo map; there is no trail. Ski or snowshoe the Gold Basin Road a short distance to the NW ridge of peak 12,271. Peak 12,271 (UTM 126253079E 4258013N) is your objective. This will be the toughest part of the climb, getting on the steep ridge and to Peak 12,271. I would advise and ice axe and crampons. Make sure to stay on a ridge to avoid any avalanche danger. Once on peak 12,271, follow the ridge south to Peale. You can also get there from Tukunikivatz, by climbing Tuk's north ridge, but this is a much tougher route. You could also climb Menenthin on the same trip, but you have to be really fit to do this in a day; just getting to Peale will be more than enough for most people on a day climb. The Gold Basin Route is the easiest route of Peale in winter and is easier than skiing or snowshoe-ing all the way up the Lasal Creek Road.
Enjoy, it's beautiful in winter.
Ice axe, crampons, snowshoes or skiis in winter.
In summer take a good pair of boots.