Heat oil in wide nonstick frying pan over medium-high heat; add beef and brown well on all sides. Meanwhile, in a 4 quart or larger electric slow cooker, combine onions, celery, and garlic; sprinkle with allspice, ginger, and pepper. In a small bowl, mix cider and molasses. Place beef on top of onion mixture; pour in cider mixture. Cover and cook at LOW until beef is very tender when pierced (9 1/2 10 hours).
Lift roast to a warm platter and keep warm. Skim and discard fat from cooking liquid, if necessary; then blend in cornstarch (blended in cold water). Increase heat to HIGH; cover and cook, stirring 2 or 3 times, until sauce is thickened (about 15 more minutes). Season to aste with salt.
To serve, remove and discard string from beef, then slice across grain. Spoon some of the sauce over meat; sprinkle with parsley. Serve remaining sauce in gravy pitcher or bowl.