Palavecino started his career with Rosario Central, he was moved into their senior squad by manager Paulo Ferrari during the 2018–19 campaign.[1][2][4] He made his senior bow on 26 February 2019 as the club were eliminated from the Copa Argentina by Sol de Mayo of Torneo Federal A, with the midfielder playing eighty-four minutes.[1] Just over two weeks later, on 13 March, Palavecino made his Copa Libertadores debut against Universidad Católica; having previously been an unused substitute for the matchday one group stage encounter with Grêmio.[1][5] In the succeeding July, Palavecino headed to Cyprus to join second tier team ASIL.[6][7]
After fourteen appearances and two goals, against Karmiotissa and Aris Limassol respectively, Palavecino returned to his homeland with Primera B Nacional's Gimnasia y Esgrima in September 2020.[8] He scored once, versus Santamarina, in five matches for El Lobo Jujeño.[1] On 17 February 2021, Palavecino returned to the Primera División with Patronato on a free transfer; penning terms until the succeeding December.[9] He debuted in a home defeat to Independiente on 21 February.[1]