Born on May 31, 1822, Edward Parrish was the son of Dr. Joseph and Susanna (Cox) Parrish, Philadelphia Quakers. In 1845 he married Margaret Shreve Hunt. Trained as a pharmacist, he operated a pharmacy in Philadelphia with his brother, Dillwyn, and taught at the Philadelphia College of Pharmacy. Edward Parrish was active on the committee of members of New York, Philadelphia, and Baltimore Yearly Meetings (Hicksite) which sought subscriptions for the founding of a college, and he was a member of the Board of the Friends Education Association which adopted a constitution in December 1862. Swarthmore College was incorporated in 1864, and Edward Parrish was one of its primary fundraisers. He was elected president of the College in May 1865, and raising money for the fledgling college continued to be one of his most important jobs. The College finally opened in the fall of 1869 in a large stone building known then as "The College" and now as Parrish Hall.

In addition to serving as president of the college, Parrish was also a professor of ethics, chemistry and the physical sciences. His liberal ideas on the discipline of students frequently clashed with some of the more conservative members on the Board of Managers who believed that the College should foremost provide a "guarded" education for Quaker youth. It is believed that the differences in philosophy of discipline and of the mission of the College are what caused Parrish to resign from the presidency in February 1871. After his resignation, the Board assumed most of the executive powers and appointed as president Professor Edward H. Magill.

Edward Parrish was active in a number of Quaker social concerns, and in 1872 he was appointed to a Commission to negotiate a treaty with the Kiowa and Comanche tribes. While on this mission to Fort Sill in the Indian Territory, Edward Parrish died of malaria at age 51 on September 9, 1872.