Stir the warm milk, yeast, and sugar together and let stand for 5 minutes. Add the flour, eggs, salt, and parsley, then mix into a soft dough.
Knead until smooth and elastic, about 6 minutes. Fold in the bread cubes, cover with a clean towel, and let rise until doubled, about 1 hour.
Shape:
Turn the dough onto a lightly floured surface and divide it into two equal pieces. Shape each into a compact log about 2 inches thick.
Cover the logs and let them rest for 10 minutes while a large stockpot of salted water comes to a gentle boil.
Cook and slice:
Lower the logs into the gently boiling water. Cover and simmer for 10 minutes, turn them carefully, then cook for 10 to 15 minutes more until firm and cooked through.
Lift the dumplings out with a slotted spoon. Pierce each log a few times to release steam, rest for 5 minutes, then slice into rounds with a serrated knife or dumpling thread.
Notes
Keep the water at a gentle simmer rather than a vigorous boil so the dumplings hold their shape.
Steam chilled slices for a few minutes to reheat them without drying them out.