Narro was born in Palma de Mallorca, Balearic Islands, and represented RCD Mallorca and SD La Salle before joining Villarreal CF‘s youth setup in January 2017, for a €40,000 fee. On 14 July 2018, after finishing his formation, he was assigned to farm teamCD Roda in Tercera División.
Narro made his senior debut on 26 August 2018 by starting in a 2–1 home win over Paterna CF, and scored his first goal on 25 November by netting the opener in a 2–2 away draw against Orihuela CF. After finishing the campaign with three goals in 35 appearances, he returned to the Yellow Submarine and was included in the C-team also in the fourth tier.
Narro spent the 2020–21 season with the reserves in Segunda División B, before departing the club on 31 July 2021. Three days later, he moved to another reserve team, Real Valladolid Promesas in Primera División RFEF on a one-year contract.