Parejo curled a free kick past goalkeeper Asier Riesgo two minutes after the break and Soldado scored in the fourth minute of added time to give the 2008 winners the upper hand ahead of next week's second leg in Valencia.
The victory at the Reyno de Navarra stadium in Pamplona made it two wins out of two for Valencia's new coach Ernesto Valverde, both against Osasuna who they also beat 1-0 in La Liga at the same venue in his debut on Saturday.
Valverde replaced Mauricio Pellegrino, sacked despite leading Valencia into the last 16 of the Champions League after they made an erratic start to their La Liga campaign.
King's Cup holders Barcelona play their last 16, first leg match at second-division Cordoba on Wednesday before 2011 winners Real Madrid visit Celta Vigo and Atletico Madrid host city rivals Getafe.
Barca's World Player of the Year Lionel Messi is in the squad for the trip to Andalusia and if the Argentine plays he will have a chance to extend his record goals tally of 86 for a calendar year.
Barca and Real are on course to meet in the last four while Sevilla face a possible semi-final with Atletico, who they beat in the 2010 final.
Sevilla play at Real Mallorca on Wednesday.
- Sports & Recreation
- La Liga
- Ernesto Valverde