Cika wannan magana: A 1987, Palmer ya yi magana da kocin McLaren Ron Dennis game da zama direban No. 2 na ƙungiyar don ninka Gasar Cin Kofin Duniya Alain Prost. Dennis a karshe ya sanya hannu kan Stefan Johansson, kuma Palmer ya shiga Tyrrell mako guda kafin tseren bude kakar a
Ƙasar Brazil. Ko da yake ya wuce ta da turbocharged fafatawa a gasa, Tyrrell ta halitta-aspirated Cosworth-powered mota tabbatar da abin dogara, kuma shi ne nimble a tighter circuits. Palmer ya sami maki a gasar a tsere uku, kuma a gasar Grand Prix ta Ostiraliya ne ya kammala aikinsa mafi kyau a matsayi na hudu. Ya kuma lashe Jim Clark Cup, gasar zakarun Turai ga direbobi na motocin da aka saba da su. Ya zauna tare da Tyrrell na tsawon shekaru biyu masu zuwa, lokacin da mafi kyawun sakamakonsa ya kasance sau biyu na 5 da kuma sau uku na 6. A ƙarshen 1989 ya sanya hannu a matsayin direban gwajin McLaren.