bus transport/ bus transportation (transportation)
paint tin/ paint can (tin/can varies by region)
TV aerial/ TV antenna (antenna or aerial antenna depending on circumstance)
mail / post (as a noun and verb) -- We mail a letter using the US Postal Sevice, the Brits post a letter using Royal Mail.
graduate school/ postgraduate school -- both depending on circcumstance.
thermos/ flask -- both ; a thermos is insulated, a flask usually isn't, & as said above, usually refers to alcohol
parcel/ package -- depends on circcumstance . Officially services are often refered to as parcel services, but we'd say "There is a package on the porch for you."
pupil/student ---- both depending on circ.
skipping rope/ jumping rope (the sport itself) -- both
kindergarten/ nursery school -- both BUT nursery school is before kindergarden and is not required by law; kindergarden is the year before 1st grade and is required by law.