Formula 2 driver Juan Manuel Correa - injured in the accident that killed French driver Anthoine Hubert - has been placed in an induced coma.
Correa, 20, is "critical but stable" in a London hospital after complications following the crash, say his parents.
The Ecuador-born American suffered a spinal injury and broken legs in the crash at Spa, in Belgium, on 31 August.
Juan Carlos and Maria Correa said they believe his "tremendous fighting spirit" will help him recover.
"We are confident that our son will surprise us like he always does," they said in a statement.
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- Hubert killed in Belgium crash
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Correa had a four-hour operation in Liege, but on arrival in London was diagnosed with acute respiratory distress syndrome.
"This is an injury considered common in high-impact accidents such as this one. Unfortunately, this injury resulted in Juan Manuel falling into acute respiratory failure," said the Correas.
"Juan Manuel is currently in an intensive care unit that specialises in respiratory injuries. At this point of time he is an in induced state of unconsciousness and under ECMO support."
ECMO (extracorporeal membrane oxygenation) is similar to equipment used during a heart-lung bypass operation,
Hubert, who was 22, suffered a huge impact from Correa's car, which was travelling at about 170mph, at the Spa-Francorchamps circuit.
The Frenchman was taken to the medical centre following the incident, and died shortly after.
Correa, who grew up in Miami and lives in Barcelona, has been racing this season with the Sauber junior team and is the development driver for Alfa Romeo Racing's F1 team.
"We wish to thank everyone within and outside the motorsports community for the wonderful and caring get well wishes that we have received, as well as messages for a speedy recovery," said his parents.
"We are confident that Juan Manuel will review them all himself once he is back in charge of his social media accounts."
They said the update had been issued in honour of the "straight-to-the-point and honesty" that characterises their son.
"At this time, we kindly request that our privacy and space be respected. As a family, we need to pull together and be 100% there for Juan Manuel," they added.
"Maria and I wish to take this opportunity to give our condolences to the Hubert family for their loss. Our hearts are broken, and we can only imagine the pain this loss has brought them."