A judge ruled that the couple must be held while the court considers whether to grant Britain an extradition request, the National Court in Madrid said.
Ashya's parents Brett and Naghmeh told the judge they do not want to return to the UK.
The boy's family took him from Southampton General Hospital last Thursday and travelled on a ferry to France with him and his six siblings before heading to the Costa del Sol in southern Spain.
Mr King, 51, and Mrs King, 45, were arrested on Saturday night in Velez-Malaga by Spanish police.
Ashya is being treated in a Spanish hospital and his grandmother and his brother have both criticised the way his parents are being treated.
Ashya's brother, Naveed, 20, said: "We're not allowed to go and see Ashya at all.
"There are police standing outside his hospital room. We are not allowed to go and see him. We have tried to call the hospital but they are not revealing any information at all to us.
"Including taking care of all the kids, we're having to do the research ourselves to find out information. We're not getting anyone knocking on our door to tell us everything is OK. We've got lots of support from friends and family out here.
"My mum was by his (Ashya's) side for the whole month that he was in hospital so for him to now suddenly not be with anyone of the family - and because he can't really move much of his body we kept him entertained, we played games with him, we make sure that he was always happy ... his health might actually deteriorate because he can't be entertained and be happy.
"We wanted the best for Ashya and for us to know that now they've taken him away from us and maybe given him treatment that may not be best for him, it's quite heartbreaking."
Naveed posted a video on YouTube yesterday to disprove the idea that the youngster had been neglected. He showed special food and medical equipment bought for the drive to Spain, and said they also had a new £1600 wheelchair for Ashya.
Naveed accused doctors in Southampton of not listening to his father, despite his hours of medical research on the internet.
He said: "He did constant research to find out information which could help Ashya which the doctors denied. They did not want to hear about his research as they did not believe any of his information that was being given to him, saying that the internet could not be trusted, whilst the internet gave him information that the doctors would not give him."
Ashya's grandmother Patricia King said she spoke to her elder grandson Danny for the first time this morning. She said: "Danny said they have taken his parents to Madrid and are giving them a terrible run-around. They want their own solicitor but are being told they have to have one the Spanish have provided. The whole thing is a huge injustice.
"They are still not allowed to see Ashya, which is shocking. It's the worst thing of all."
The couple's decision to take their son out of hospital has generated heated debate on social media about their right to decide what treatment Ashya receives.
According to reports, they travelled to Spain to sell a holiday home to pay for proton beam therapy, not available through the NHS. Their son, who is suffering from a stage-four brain tumour, is being cared for at the Materno-Infantil hospital in Malaga.
British police have travelled to Spain to question the couple and defended their decision to request a European arrest warrant for them on suspicion of neglect.
Assistant Chief Constable Chris Shead, of Hampshire Constabulary, said he was aware the police's approach had created a "significant amount of debate" but he would rather be criticised for "being proactive" than "potentially having to explain why a child has lost his life".
Simon Hayes, police and crime commissioner for Hampshire and the Isle of Wight, said: ""Hampshire Constabulary's role, as in many other cases, was to safeguard the interests of a very vulnerable sick, young child.
"Medical professionals at Southampton General Hospital had advised that Ashya's life was in 'grave danger', so the constabulary understandably acted to protect a very vulnerable sick young boy. If they had ignored professional medical advice, then they would have been negligent."