Allen attended Fayetteville High School in Fayetteville, Arkansas. As a senior, he threw for 3,408 yards and 38 touchdowns. He was ranked by Rivals.com as the fifth best pro-style quarterback recruit.
Allen was redshirted as a freshman in 2011. In 2012, he was the backup to Tyler Wilson. He appeared in five games and started one game after Wilson missed a game because of injury. He finished the season completing 21 of 49 passes for 186 yards with one touchdown and three interceptions.
In 2013, Allen took over as the starting quarterback. He passed for 1,552 yards, completing 49.6 percent of his throws, and threw for 13 touchdowns and 10 interceptions for a team that finished 3-9 under first year head coach Bret Bielema.
As a junior in 2014, Allen had 2,285 passing yards, 20 touchdowns and five interceptions. He was the offensive MVP of the 2014 Texas Bowl, a 31-7 victory against the Texas Longhorns. Allen threw two touchdowns in the game, and Arkansas finished with a 7-6 record.
During his senior year in 2015, Allen threw a then school record six touchdown passes, and ran in the game winning 2 point conversion in a 53-52 overtime win over Ole Miss. He broke the record two weeks later with seven touchdowns in a 51-50 loss to Mississippi State. He also broke Arkansas' record for career passing touchdowns against Mississippi State. He finished the year with 3,440 passing yards, 30 touchdowns, and eight interceptions for a team that finished 8-5, and beat Kansas State in the 2016 Liberty Bowl, 45-23.
Allen finished his career with 7,463 passing yards, 64 passing touchdowns and 26 interceptions.
Allen was drafted by the Jacksonville Jaguars in the sixth round (201st overall) in the 2016 NFL Draft. On May 5, 2016, Allen signed a 4-year $2.48 million contract, which included a $147,687 signing bonus. Allen was the Jaguars third-string quarterback his rookie season behind Blake Bortles and Chad Henne. He was waived by the Jaguars on September 3, 2017, following final roster cuts for the 2017 season.
On September 4, 2017, Allen was claimed off waivers by the Los Angeles Rams.
His father, Bobby Allen, is the Razorbacks’ director of high school and NFL relations. He has a younger brother, Austin Allen, who will be a senior quarterback at Arkansas in 2017, and who became the starting quarterback for the Razorbacks after Brandon completed his college eligibility, and was drafted in 2016. Allen graduated in 2014 with a degree in recreation and sport management.