The one-hour afternoon session at Jerez saw little in the way of overall lap time improvements until the closing stages, when FP2 exploded into a frantic time attack.
Espargaro was the best of the bunch with a 1m36.708s from Aprilia team-mate Maverick Vinales, while FP1 pacesetter Dani Pedrosa was third overall.
As disastrous afternoon for last year’s Spanish GP winner Bagnaia, Quartararo – who has four Jerez MotoGP poles to his name – and championship leader Marco Bezzecchi saw all three fail to get into the top 10 and will face a volatile Q1 session on Saturday morning.
At least one of those riders will start both the sprint and the grand prix from the fifth row at best.
After half an hour of running in FP2, only Marc Marquez’s injury stand-in Iker Lecuona had made an gains on their FP1 best.
But Lecuona remained last on the timesheets and suffered a crash 12 minutes into the hour-long session at the final corner.
This would eventually bring the red flag out 10 minutes later as his Honda had punctured the protective air fence lining the edge of the track at Turn 13.
After a brief pause, the session got back under way with 37 minutes and 41 seconds left on the clock – though it wouldn’t be until the final 13 minutes that lap times started to improve significantly.
The first major mover was Vinales, who shot up to third on the combined standings with a 1m36.899s.
With 40s of FP2 remaining, Vinales would finally depose KTM wildcard Pedrosa’s best from FP1 with a 1m36.710s.
However, it wouldn’t be enough to keep the Spaniard at the top of the timesheets as Espargaro snatched the lead away from Vinales by 0.002s.
KTM’s Jack Miller was third on the session timesheet, but fifth overall, with Johann Zarco (Pramac Ducati) heading RNF Aprilia’s Miguel Oliveira.
Alex Marquez was sixth in FP2 on his Gresini Ducati ahead of VR46’s Bezzecchi and Ducati’s Bagnaia, with Pedrosa and the second VR46 Ducati bike of Luca Marini rounding out the top 10.
However, front-end struggles throughout FP2 for Bagnaia meant his 1m37.233s – which was only 0.525s off the pace – was only good enough for 13th overall behind Bezzecchi, who suffered an engine blowout late on.
Quartararo fared even worse as he was only 16th on his factory Yamaha ahead of team-mate Franco Morbidelli, with a twice-crashing Joan Mir 18th for Honda.
At present, neither factory Ducati rider is in Q2 as Enea Bastianini continued to struggle with his injured shoulder down in 21st on the combined order at the end of FP2.
As well as both factory Aprilias, both Pramac riders, Miller and Pedrosa on his wildcard outing, Oliveira, Alex Marquez, Takaaki Nakagami and Marini all bagged a direct Q2 spot for Saturday morning.
Spanish GP – FP2 results:
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Tyisha Noren