Place the chicken in a 4‑quart slow cooker and add 1 to 2 cups of water or chicken broth.
Cover and cook on low for 4–6 hours (or 3–4 hours for chicken breasts) until the chicken is cooked through and tender.
Remove the chicken with a slotted spoon and drain the liquid from the slow cooker; discard any visible fat.
Transfer the chicken to a cutting board and shred it with two forks or meat claws.
Return the shredded chicken to the slow cooker. Add 1/2 to 1 cup Caesar dressing, 1/2 cup shredded Parmesan, 1/4 cup chopped parsley, and 1/2 teaspoon ground pepper.
Stir well to combine, cover, and cook on high for 30 minutes, or until the mixture is heated through.
Assemble sandwiches by spooning about 1/4 cup of the chicken mixture onto each slider bun (use more for regular buns), then top with extra Parmesan and shredded romaine lettuce.
Instant Pot alternative: add chicken and about 1 cup water or broth to the Instant Pot, cook at pressure for 8 minutes, allow natural release, drain, then proceed with shredding and Step 5.