Bardock ends up battling an ancestor of Frieza's, Lord Chilled, which triggers the Super Saiyan transformation in Bardock. This is where things get a little weird, as Lord Chilled goes on to warn ...

Super Saiyan 3 transformation not used in DBS due to massive Ki drain, explains why Vegeta opted out. Vegeta's growth seen in various transformations over the years, finally achieving Super Saiyan 3.