In a 4-quart saucier, combine beans and 3 quarts (2.8L) salted water. Bring to a boil over high heat. Remove from heat, cover, and let soak for 1 hour. Drain and rinse beans; set aside.

Meanwhile, in a large Dutch oven, heat oil over medium heat until shimmering. Add onion, carrots, and celery, and cook, stirring occasionally, until softened, about 10 minutes. Add garlic and cook, stirring frequently, until fragrant, about 60 seconds. Transfer vegetables to a small heatproof bowl; set aside.

In the same Dutch oven, add remaining 2 ½ quarts (1.9L) water, along with thyme, parsley, bay leaf, and ham hocks. Bring to a boil over high heat, then lower heat to maintain a gentle simmer. Cover and cook until ham hocks are tender, about 1 hour.

Stir in beans and bring to a simmer over medium high heat. Reduce to medium-low and cook, covered, until beans are completely tender and meat is falling off the bones, about 2 hours, stirring halfway through. Discard thyme, parsley, and bay leaf. Using tongs, transfer ham hocks to a cutting board and set aside until cool enough to handle, about 20 minutes. Using a slotted spoon, transfer 2 ½ cups beans to a medium heatproof bowl; set aside.

Once ham hocks are cool, roughly chop meat using a sharp knife; discard skin, bones, and fat. Set aside.

Using an immersion blender, blend soup until smooth (if using a countertop blender, blend in batches, if necessary, and start blending at low speed before increasing to high, then return puréed soup to the pot). Add reserved beans, meat, vegetables back to pot, stirring to combine. Stir in 1 to 2 cups of stock or water to reach desired consistency. Return to a gentle simmer over medium-low heat, then cook, stirring occasionally, until vegetables are tender and ham is heated through, about 20 minutes. Season to taste with salt and pepper.

Divide soup among warmed bowls, sprinkle with parsley and pepper, and serve.