Prepare the Salted Caramel:
Place sugar in an even layer in a medium-sized saucepan over medium-high heat. Allow it to melt without stirring until it begins to liquefy. Stir with a whisk as it melts until it reaches a deep amber color (around 350°F), using a thermometer if available.
Carefully add butter cubes all at once, whisking until smooth. The mixture will bubble up, so take caution. Switch to a wooden spoon if needed.
Remove from heat and slowly pour in the heavy cream, stirring until fully incorporated. Stir in sea salt and let the caramel cool for 10-15 minutes.
Assemble the Parfaits:
In a large bowl, whisk together the pudding mix, pumpkin pie spice, and milk until thickened, about 2 minutes. Fold in the pumpkin puree and set aside.
In 8 small jars or serving glasses, layer about 2 tablespoons of chopped gingersnaps at the base.
Add a layer of pumpkin pudding, followed by a layer of cream cheese frosting. Drizzle with a spoonful of salted caramel.
Repeat the layers until all ingredients are used, finishing with a drizzle of salted caramel and extra gingersnap crumbles on top.
Chill parfaits in the refrigerator for at least 2 hours before serving.
Storage Tip:
Extra salted caramel can be stored in a sealed jar in the refrigerator for up to 2 weeks. Reheat gently before using.