I live on the west coast (Oregon). My parents live on the east coast
(Connecticut). They have several pieces of furniture they want to gift
me but shipping by conventional means (USPS, FedEx, etc) would cost a
small fortune. I've been told that some moving companies will take
items across the country when their vans are partly full, but I'm not
sure how to identify these or other opportunities. Please recommend my
best options. I expect to do this more than once in the coming years,
as my parents downsize back east while my family in Oregon grows.

Moving companies are your answer.  My brother in law owns a nationwide
agency: Paul Arpin of Richmond / Stewart Moving and Storage
(www.stewartmovingandstorage.com)

They, or any other reputable nationwide moving company can and will
move your funiture.  It may take a bit longer, but is handled
professionally.

Good Luck

Thank you very much. I called Stewart Moving and Storage and they
recommended an Arpin agent in Pennsylvania, where the furniture is
located. I had assumed that my needs would be too small for large
moving companies but apparently not.