Pelham spent the 2016 season with the Spokane Indians of the Low-A Northwest League, posting a 0–6 record, 6.16 ERA, 50 strikeouts, and 38 innings pitched in 16 games (7 starts).

[5] Pelham spent the 2017 season with the Hickory Crawdads of the Single-A South Atlantic League, posting a 4–2 record, 3.18 ERA, 75 strikeouts, and 62+1⁄3 innings over 37 games.

Pelham began the 2018 season with the Down East Wood Ducks of the High-A Carolina League, posting a 0–0 record, 1.95 ERA, 34 strikeouts, and 27+2⁄3 innings in 23 games.

[6] He was promoted to the Frisco RoughRiders of the Double-A Texas League on June 22, and posted a 2–0 record, 6.16 ERA, 19 strikeouts, and 19 innings in 24 games.

[10] In 2019, Pelham split the season between Frisco and the Nashville Sounds of the Triple-A Pacific Coast League,[11] combining to go 1–4 with a 11.97 ERA over 32+1⁄3 innings.

In 2022, Pelham made 29 appearances split between the Double-A Tennessee Smokies and the Triple-A Iowa Cubs, posting a cumulative 2–1 record and 4.35 ERA with 41 strikeouts and 2 saves in 41+1⁄3 innings pitched.

[19] In 10 appearances split between Salt Lake and the rookie–level Arizona Complex League Angels, he compiled a 1.80 ERA with 12 strikeouts across 10 innings of work.