Heat the sesame oil in a large pot over medium heat. Add the onion, garlic, and ginger and cook for 3 minutes until the onion softens and the mixture smells fragrant.
Stir in the daikon pieces and parsnip rounds. Cook for 2 minutes to coat the vegetables lightly with the oil and aromatics.
Pour the vegetable broth into the pot. Bring the mixture to a boil, then lower the heat and simmer uncovered for 20 minutes until the daikon and parsnips are tender when pierced with a fork.
Place the miso paste in a small bowl and ladle in about 1/2 cup of the hot broth. Whisk until smooth, then stir the mixture back into the pot.
Heat the stew for 2 more minutes on low heat without letting it boil. Remove from heat and ladle into bowls.
Top each serving with sliced green onions and toasted sesame seeds.