Palazón was born in Cieza, Murcia, and made his senior debut with CD Cieza at the age of just 15, after spending a short period at Real Madrid's La Fábrica.[1] He subsequently moved to Villarreal CF's youth setup, reaching the C-team before being released in 2013; he then returned to his first club Cieza.[1]
In 2014, Palazón joined Real Murcia and was initially assigned to the reserves in Tercera División. On 18 December of that year, he agreed to a new three-year contract with the club, being definitely promoted to the main squad in Segunda División B the following year.[2]
On 18 August 2017, Palazón moved to fellow third division side SD Ponferradina, after terminating his contract with Murcia.[3] He contributed with eight goals in 40 appearances during the 2018–19 season, as his side returned to Segunda División after a three-year absence.
Palazón made his professional debut on 18 August 2019, starting in a 3–1 away loss against Cádiz CF.[4] He scored his first goal in the category on 26 October, netting his team's second in a 3–1 away defeat of Rayo Vallecano
On 23 January 2020, Palazón signed a three-and-a-half-year contract with fellow second division side Rayo.[5]