There are far more father-son duos of position players than pitchers. Three Cy Young Award winners have had a son reach the majors. Vern Law’s son Vance was an infielder. Steve Bedrosian and Doug Drabek had sons (Cam and Kyle) reach the majors as pitchers but neither merited any Cy Young Award attention.
Clyde Wright placed sixth in 1970 Cy Young voting in the American League. His son Jaret Wright pitched well in the 1997 playoffs as a 21-year-old. He looked like he might develop into a Cy Young Award contender but never put it all together, finishing with 68 wins over 11 seasons.
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Best father-son pitching combo was Mel and Todd Stottlemyre. They combed for 302 wins (Mel 164, Todd 138). Neither ever received a Cy Young Award vote, although Mel likely would have in the 1960s if there were votes for more than just first place then.
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Players of the same name to receive Cy Young Award votes
18 Mike/Michael – Bielecki, Boddicker, Caldwell, Cuellar, Flanagan, Foltynewicz, Hampton, Krukow, Marshall, McCormick, Minor, Moore, Mussina, Norris, Scott, Soroka, Witt, Fulmer
17 John/Jon/Johnny – Axford, Burkett, Candelaria, Denny, Franco, Hiller, Lackey, Montefusco, Smiley, Smoltz, Tudor, Wetteland, Cueto, Garland, Lester, Lieber, Matlack. Add Jacks to the John list and they take the lead.
12 Jim/James and Dave/David