Bottom club Peterhead continued to press after the break and Wyness drew another save from Bullock.
And a draw was confirmed after Peter McVitie spurned a late chance to win the match for the hosts, who are yet to record a league victory this season.
Peterhead: Bateman, Gethans, Strachan, Donald, MacDonald, Stuart Smith, Moore (Scott Ross 72), McVitie, Anderson (Sharp 10), Bavidge, Wyness. Subs Not Used: Jarvie, Hepburn, McClaughlan.
Livingston: Bullock, Sinclair (Kyle Jacobs 83), Craig Barr, O'Byrne, Watson, Talbot, Keaghan Jacobs, Fox, Bobby Barr, Russell (De Vita 74), Winters. Subs Not Used: McDowall, McNulty, Conway.
Booked: Fox, Craig Barr, O'Byrne, Bobby Barr, Kyle Jacobs.
Att: 431
Ref: J Beaton
Source: BBC Sport
Source: BBC Sport