Why is Psoriasis treated with Cortisone, when stress (which releases Cortisol) is known to aggravate the condition?
For treatment reasons cortisol is named Hydrocortisone (probably to avoid problems with the similar names) and this is often only referenced to as cortisone (although this is not correct). Cortisol (Hydrocortisone) is the active form, while cortison is not. See here for some more details.

Glucocorticoids (like Cortisol or Prednisolone) suppress acute inflammation and also have immuno modulative (downregulate) or suppressive function. Since the immune system causes the problems in Psoriasis it is downregulated to control the outcome of the disease.

First it sounds counterintuitive to use Glucocorticoids for the treatment of psoriasis as this hormone is released during stress (in this case to mobilize energy resources). However, there is a study which looked precisely at the cortisol levels in psoriasis patients and also recorded their stress levels. 

What they found was interesting: First they could correlate the peaks in the stress levels with a psoriasis attack about four weeks later. Then they found that the Cortisol levels of the patients where significantly lower in stres situations than for healthy control persons. Additionally they found that patients which had permenant high stress levels also showed significantly lower permanent Cortisol levels. So here the stress seems to negatively influence the cortisol levels which then prepares the way for psoriasis attacks. Using high levels of topically administered Cortisol seems to reverse this effect.
You can find the study here: 


How stress gets under the skin: cortisol and stress reactivity in
psoriasis.