Lehlohonolo Majoro scored one and created the other two to give the Glamour Boys a comfortable win against their Soweto neighbours.
Amakhosi were full value for their win as they were the better side throughout the match, with Swallows failing to create chances and hardly threatening Itumeleng Khune.
Majoro, who came on for Kingston Nkhatha in the 36 minute, opened the scoring in the 43rd minute after receiving a long pass from Josta Dladla before Parker double the lead in the second half.
Lebese made it 3-0 in the 87th minute after he combined well with Majoro, who had an outstanding game.
Coach Stuart Baxter started with new signings Morgan Gould, Eric Mathoho, Siboniso Gaxa, Siyabonga Nkosi and Kingston Nkhatha and they all had an impressive debut.
Chiefs: Khune, Gaxa (Moleko), Mathoho, Gould, Ritchie, Tshabalala (Lebese), Dladla (Nale), Letsholonyane (Baloyi), Nkosi, Parker (Motaung Junior), Nkhatha (Majoro)
Swallows: Mathibe, Mapemba, Mungomeni, Da Costa, Lekgothoane, Sibaya, Tsutsulupa, Chabangu, Chenene (Lwazi), Makhanya (Obada), Maleka (Nomvethe)