Cover the chickpeas with 4 cups water in a large pot with 1 teaspoon baking soda and refrigerate 8 hours or up to overnight to soak.
Strain the chickpeas and discard the soaking liquid. Rinse the chickpeas and return them to the pot. Add the remaining teaspoon baking soda and water to cover by about 3 inches over the chickpeas. Bring to a boil over high heat, then turn the heat down to a simmer and cook, skimming off and discarding any foam that rises to the surface. Cook the chickpeas until tender but still hold their shape, about 30 minutes. Remove 1 cup of chickpeas from the pot with a slotted spoon. Drain well and reserve on a paper towel-lined plate to dry. Continue cooking the remaining chickpeas until they are mostly intact and soft and creamy (not crumbly) when pressed between a thumb and forefinger, about 10 minutes more. If the liquid boils off too much while cooking, add more water to cover and continue to cook.
Meanwhile, make the parsley oil. Bring a small pot of water to a boil. Add the parsley leaves and cook 10 seconds, then drain. Immediately run the parsley under very cold water to cool it completely, 30 seconds to 1 minute. Squeeze out the excess liquid and then transfer to a paper towel-lined plate to dry.
When the chickpeas are done cooking, strain them and discard the liquid. Let the chickpeas sit in the strainer to drain and dry out for 5 minutes. Put the chickpeas in the bowl of a food processor along with the tahini, lemon juice, garlic and 1 3/4 teaspoons salt and process until the mixture is a paste, scraping down the bowl as necessary. With the motor running, slowly pour in 1/4 cup of the ice water and continue to process, adding more water by the tablespoon as necessary, until the mixture is smooth and light, about 5 minutes. The hummus should be slightly loose because it will firm up as it sits.
Transfer the hummus to a shallow bowl and cover by lightly pressing plastic wrap directly onto the surface; set aside until ready to serve. (The hummus can also be made a day ahead and refrigerated. Bring to room temperature for 30 minutes before serving.)
Put the cooked parsley leaves in a blender along with 1/2 cup olive oil and puree until the parsley is fine and the oil green, about 5 minutes. Transfer the parsley oil to a small bowl and cover with plastic wrap pressed directly against the surface. Parsley oil can be used right away, or held at room temperature up to 6 hours
Just before serving, fry the chickpeas. Heat the remaining 1/2 cup oil in a medium skillet over medium-high heat for about 2 minutes. Pat the reserved chickpeas dry with a clean towel and carefully add them to the skillet. (The oil will bubble up for a few seconds.) Fry, shaking the pan occasionally, until browned and crisp, 7 to 9 minutes. Transfer the chickpeas to a paper towel-lined plate to drain. Immediately sprinkle with the lemon zest, cayenne, cumin and some salt; toss to coat.
Remove the plastic wrap from the reserved hummus. Make a shallow circular indentation with the back of a spoon in the center of the hummus. Mound the fried chickpeas in the center of the hummus. Drizzle with the parsley oil, spooning most of it around the edges of the hummus.
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