Father-President is a fairly common title in Catholic institutions where the chief executive is a priest.
The Spanish missions in North America were a "co-venture," with the Catholic Church seeking converts and the Spanish Crown seeking to "Hispanize" the native population. The former supplied the manpower for converting and educating the Indians, the latter the funding for salaries, supplies, and military protection; the missionaries. After the suppression of the Jesuits, the Spanish Crown gave the mission franchise in the Californias to Franciscans, who assigned the Californias to the Apostolic College of San Fernando, one of their missionary colleges in Mexico City. The College would have had authority over its priests. The superior in California, i.e. the padre presidente, would have been elected, although in several cases successors were elected in advance, e.g. for Tápis to take command after Lasuén.
Zephyrin Engelhardt's The Franciscans in California says "It had long been the intention to found a series of new missions, each equidistant from two of the old establishments." Most of the general locations were thus set after a series of surveys in the 1790s.
The New Catholic Encyclopedia article on California covers the relationships. After 1783, the father-president was in contact with the Bishop of Sonora, who named some of the father-presidents of the missions as vicar forane ("rural dean"), military vicar, and even vicar general. This provided them ordinary authority, for example, to conduct confirmations, in the absence of the bishop. This situation persisted until 1840. Still, these powers were conferred, and not inherent in the position of father-president, and the selection of father-president remained with the College of San Fernando, not with the episcopate.
In 1812, with Mexican independence, the office was split, with a father-president responsible for strictly ecclesiastical matters and a commissary-prefect (comisario-prefecto) for the missions' business affairs and temporal relations. The first commissary-prefects were appointed by the Franciscan order's Commissary General for the Indies in Spain, but later elected locally, de facto. (The office was vacant from 1830-1836.) In 1833, with Governor Figueroa seeking to rid Mexico of Spanish members of religious orders, the missions in northern California were transferred to the Apostolic College of Our Lady of Guadalupe in Zacatecas.
The New Catholic Encyclopedia gives the following lists:
Father-Presidents: Junipero Serra (1768-1784), Fermín Francisco de Lasuén (1785–1803), Estevan Tapis (1803–12), José Señan (1812–15, 1820–23), Mariano Payeras (1815–20), Narciso Durán (1824–27, 1830–36), José Bernardo Sánchez (1827–30), and José Joaquín Jimeno (1839–53)
Francisco Palóu served as acting president between Serra and Lasuén.
Commissary-Prefects: Vicente Francisco de Sarriá (1812–18, 1824–30), Mariano Payeras (1818–23), Narciso Durán (1836–46), and José Joaquín Jimeno (1846–53)
Commissary-Prefects in northern California: Francisco GarcÍa Diego y Moreno (1833-1840)
Father-Presidents in northern California: Rafael Moreno (1833-?)