Knockalla was euthanised after being injured when it was frightened by a vehicle on 30 September 2024.
It moved out into the middle of the road as a result and was hit, fracturing its leg.
The rider, a 17-year-old girl, sustained minor injuries and bruising.
The incident spurred calls from organisations like the British Horse Society for greater driver awareness and care, urging drivers to slow down to a maximum of 10 mph (16 kmph) and pass horses with a wide berth.
Lambourn Parish Council launched a Dead Slow campaign to improve safety for horses and riders on roads in the area.
Mr Dillon’s bill also aims to make equestrian safety part of the driving theory test and ensure equestrian awareness is taught in driver education.
He said: “Horses and riders are part of the fabric of life here in West Berkshire, but too often our roads are unsafe for them.
“The tragedy in Lambourn is a heartbreaking reminder of how high the stakes are.”
A Presentation Bill is a type of Private Members Bill that is introduced without debate in the House of Commons.
They rarely become law without government support, but Mr Dillon said the bill was “putting equestrian safety firmly on the agenda in Westminster” and was an important step in keeping pressure on ministers to act.
Highway Code guidance currently states drivers should slow to 10mph and leave at least two metres when passing horses.