Heat a large soup pot or dutch oven on medium-high heat. Add a drizzle of oil and sear the ham hock, 3 minutes per side. Remove from pot and set aside.
Add onion, carrot, celery, and jalapeno, and cook until soft, 4-5 minutes. Add garlic and cook an additional 1 minute.
Place ham hock back into the pot and add about 2/3 of the split peas. Cover with chicken stock and bring to a boil.
Reduce heat to low and cover. Simmer, stirring occasionally, for 1 hour.
Add the remaining split peas and cook for another 30 minutes.
Remove from heat and add the heavy cream, stirring to combine. If you prefer a smooth texture, add soup to blender or food processor and blend until smooth.
If using extra ham pieces, mix those into the soup. Garnish with chopped bacon and parsley. Serve warm.