The hike depends on where the park is - in the early season when it is on the mile it is about a 45 minute hike up the climbers trail.  Later, when the path is free of snow, it only takes 30 minutes or so.  When they move the park up to the lower palmer, it can take up to 1 hour 20 minutes.
It is sort of a brutal hike, though, because it is all up hill and each step takes you higher with no respite.