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Heat oven to 200C/180C fan/gas 6.
Heat the oil in a large frying pan over a high heat until hot. Season the lamb racks allover, then sear them in the pan on all sides until golden.
Tip the potatoes, garlic and rosemary into a large roasting tin and mix with the last of the oil and some seasoning. Sit the lamb racks on top and roast for 20 mins for medium-rare meat or 30 mins for medium. Remove the lamb from the tin to rest and mix the olives and vinegar into the potatoes; return to the oven for 5 mins to warm through.
To serve, carve the lamb into cutlets and serve with the potatoes, scattered with the parsley.