Peel the white asparagus from right below the head to the end. Once all the asparagus has been peeled, cut or break off the ends about one inch up (2.5 cm).
Cut the morels into halves lengthwise or into smaller pieces if the morels are big.
Clean the morels by brushing them with a soft brush. If they are very dirty, rinse them under running water.
On a frying pan over medium heat, melt the butter. Reduce the heat to low and add the morels. Sauté for about 5 minutes. It is extremely important not to sauté the morels over high or medium-high heat. They will lose flavor, and they should also not take any color.
Turn up the heat a bit and add the heavy cream. Let it simmer for 12 minutes.
Meanwhile, bring a large pot of water to a boil. Add 1/2 teaspoon of salt to the boiling water and add the white asparagus. Boil for about 3-4 minutes depending on the thickness of the asparagus. Once the asparagus are done, drain and keep them in a kitchen towel.
When the morels are ready, add the finely chopped parsley to the morels and season with salt and pepper. Add a bit more heavy cream if necessary.
Divide the white asparagus and the creamy morels on 4 plates and serve immediately.