The body of Ricardo Solis was discovered partially on fire in front of a home in western Sydney in the early hours of Thursday.
Police allege the man in his 60s was murdered outside his home by his son Edicar Solis following a confrontation.
The younger man was arrested hours later and faced Penrith Local Court on Friday, sporting a bandaged hand.
His lawyer Javid Faiz told the court the alleged murderer did not want to be referred to by his last name.
Instead, the accused man wanted to use the name Ardi.
Mr Faiz asked that his client be assessed by two psychiatrists as he pushed to have the matter dealt with under mental health laws.
Solis had been released from custody in April, with material indicating he needs to see a psychiatrist for ongoing mental health issues, his lawyer told the court.
Judge Stephen Corry said the alleged murderer could receive treatment while in custody.
"Given the nature of the facts involving significant violence, I wouldn't have thought it would be appropriate for a person to be held in a mental health facility," he said.
He ordered Solis be assessed by NSW Corrective Services before the matter returns to court in September.
A police application for a court order to carry out a forensic procedure on Solis was adjourned to the same date.
Solis, who maintained a blank expression throughout the proceedings, did not apply for bail.
Outside court, Mr Faiz said the allegations against his client were serious.
"It's a tragic incident," he said.
"I have asked for this matter to be considered to be looked at through (the lens of) mental health.
"Mental health ... needs to always be considered amongst families that if there is a mental health issue, then it's always looked at before it's too late."
Solis was arrested at 6.45am on Thursday, around five hours after the alleged altercation with his father.
He was treated for minor injuries before being taken to hospital under police guard on Thursday, police said.
Solis was remanded in custody until the next court date in two months, pending any bail application.
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