This miniature portrait is unsigned, but is attributed to Thomas H Parker (1801-1851). The sitter is Professor William Tully MD (Nov 18, 1785 - Feb 28, 1859), who was professor at the Yale Medical School from 1829-1842. He also wrote medical books.
In the rear of the miniature case there is an old note which reads "Prof Tully, Yale College, bought of Avery, Springfield, Mass Aug 24 1903. Avery bought it off daughter of Mrs Tully".
There is also a note, presumably from Avery which is headed "Hare & Coolidge - Ogunquit, Maine - Antiques - authentic and guaranteed - Miniature on ivory American - of Prof William Tully of Yale, taught there 1829-41 by (attributed to) Thomas H Parker 1801-1851".
Also displayed here are close-ups of two other portraits of William Tully, a miniature painted around the time his Journal was written and a large oil painting of him in middle life. The originals of both these portraits are owned by the Yale University Art Gallery. The likeness between the oil and the miniature of him in later life can be clearly seen, especially his nose, his eyes, and his double chin.
The manuscript of William Tully's 1908-1909 Journal, written while he was a medical student at Dartmouth, was discovered at Yale University Library in 1959, but a publisher was not found until 1977. The title page is shown here. The Journal was a gift to Yale from the Misses Tully in 1896. Presumably, they retained the miniature at that time and later sold it to Avery in 1903.
For more about Tully see Yale and Medicine, 1701-1901: The Medical Institution of Yale ... 904