A 25-year-old man and a 17-year-old boy are in custody after being arrested separately on suspicion of murder, police said.
Thavisha Lakindu Peiris, 25, of Sri Lankan origin was found dead in his car on Sunday night in the Southey area of Sheffield as he was on his way to make a pizza deliver, the last order before he was to join an IT firm.
Peiris was on his last shift before starting his dream job when he died in what was an apparently random attack, police said.
Speaking from their family home in Sri Lanka, his father, Sarath Mahinda Peiris and mother, Sudarma Narangoda, said "We sent our son to the UK to study so he can have a better life. Now we are left with only a broken heart."
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"We were devastated and shocked to hear of our son's tragic end and we are still unable to comprehend that he is actually gone," they said.
Peiris was found by fellow Domino's workers who had gone looking for him after he failed to deliver his last order.
He had just secured a job as an IT consultant in the city.
His colleagues and police said he had been feeling unwell during his last shift and was going to go home early but agreed to deliver one last order.
More than 50 police staff are involved in the investigation and his family are understood to be making plans to travel to South Yorkshire.
The officer leading the inquiry, Detective Superintendent Lisa Ray, said she did not know why Peiris was killed in an apparently random attack.