[7] He was assigned to the Rookie-level Arizona League Angels, with whom Bedrosian had a 0-2 win–loss record and a 4.50 earned run average (ERA) in five games (four starts).

After starting the 2014 season with Inland Empire with 15 strikeouts in his first 20 batters faced, the Angels promoted him to the Arkansas Travelers of the Double–A Texas League in April.

[14] He opened the 2020 MLB season, which was shortened to only 60 games due to the COVID-19 pandemic, as one of three setup men to closer Hansel Robles, splitting the job with Ty Buttrey and Keynan Middleton.

[15] He was a steady arm in the bullpen through the first 10 games, with only one earned run in his first 4+1⁄3 innings, before a strained adductor muscle on his right thigh forced Bedrosian onto the injured list on August 4.

[18] Despite showing steady improvement from 2018 to 2020, with an era of 3.41 across 141 games in that time frame, the Angels outrighted Bedrosian off the 40-man roster at the end of the season.

[19] On February 16, 2021, Bedrosian signed a minor league contract with the Cincinnati Reds, a deal that included an invitation to spring training.

[20] After impressing general manager Nick Krall with 16 strikeouts and only three hits in 8+2⁄3 spring training innings, Bedrosian was promoted to the major league roster on March 29.

When Cincinnati needed to clear a spot on the 25-man roster for Sonny Gray, who was activated from the 10-day injured list on April 17, Bedrosian was designated for assignment.

[22] He cleared waivers and was assigned to the Reds' alternate training site in Louisville, but declined the move and elected free agency.

[25] After recording a 2.00 ERA in nine games for Oakland, Bedrosian was once again designated for assignment on June 29, this time as part of a larger series of roster moves to make room for infield prospect Frank Schwindel and reliever J.

[27] On July 7, 2021, Bedrosian signed his third minor league contract of the year, this time with the Philadelphia Phillies, with whom his father had won the Cy Young Award in 1987.

He made 16 appearances for the Lehigh Valley IronPigs, going 2–2 with a 2.25 ERA and striking out 20 batters in as many innings, before joining the Phillies when rosters expanded on September 1.

In doing so, the Bedrosians became the third father-son duo to play for Philadelphia, following Rubén Amaro Sr. and Jr. and Mark Leiter Sr. and Jr.[28] He pitched in 11 games for the Phillies, mostly in high-leverage situations during their ultimately unsuccessful run to capture the NL East title.