Dumbledore (and the Ministry of Magic) believed that the 'killing curse' was used on Riddle Sr. due to the lack of physical injury on the bodies.

‘The Muggle authorities were perplexed. As far as I am aware, they do not know to this day how the Riddles died, for the Avada Kedavra Curse does not usually leave any sign of damage … the exception sits before me,’ Dumbledore added, with a nod to Harry’s scar. ‘The Ministry, on the other hand, knew at once that this was a wizard’s murder. They also knew that a convicted Muggle-hater lived across the valley from the Riddle house, a Muggle-hater who had already been imprisoned once for attacking one of the murdered people.

_{^{Harry Potter and the Half-Blood Prince}}

The wand in question was searched (presumably to determine the most recent spell/s used) and seems to have agreed with the Ministry's initial diagnosis that the Riddles had been killed with the Killing Curse.

He handed over his wand, which was proved at once to have been used to kill the Riddles.

There is no where in canon that explicitly states it, but there is a passage in Goblet of Fire describing the lack of anything wrong with the bodies, other than an expression of fear.

I don't have the books in front of me, but the lack of any cause of death is what stalled the legal proceedings against Frank Bryce, the gardener.

He was released, but the suspicion followed him until his death.

Regardless, the Killing Curse is the only known magical cause of death that must be determined by ruling out all other possibilities.

I'll edit this later with the references.

The Avada Kedavra curse was used

It is stated that in various parts of the books, notably in Half-Blood Prince, that the Killing Curse does not leave physical signs of damage.

The Muggle authorities were perplexed. As far as I am aware, they do not know to this day how the Riddles died, for the Avada Kedavra Curse does not usually leave any sign of damage.

In the introduction of Goblet of Fire, which delves into the murder of the Riddles and the Muggle police initially suspecting their gardener, the Muggle police receive a forensic report detailing that there was no discernible reason as to why they died.

The police had never read an odder report. A team of doctors had examined the bodies and had concluded that none of the Riddles had been poisoned, stabbed, shot, strangled, suffocated, or (as far as they could tel ) harmed at all. In fact (the report continued, in a tone of unmistakable bewilderment), the Riddles all appeared to be in perfect health — apart from the fact that they were all dead.

While there may be other magical methods to get this done, Tom Riddle Jr wanted to frame Morfin for the murders, and most other methods for murder carried a risk of being seen as out-of-character for Morfin, who has demonstrated preference and skill for the Dark Arts, engages in openly hostile behaviour towards the Riddles and has a plausible motive for murder.

To that end, while modifying Morfin's memory might have been enough, the spell that was last used by that wand being the Killing Curse would seal the deal.

Voldermort probably used the Imperius curse on Morfin and then made Morfin use the killing curse on the Riddles, or some other type of manipulation. Dumbledore heavily implies that he has no doubt it was Voldermort that killed them. Voldermort despised his family, who were uncivilised swamp dwellers probably with a history of inbreeding.