An Oireachtas committee has recommended that traffic fines should be discounted by half if paid within 14 days.
In its submission on the new Road Traffic Bill, the Joint Oireachtas Committee on Environment, Transport, Culture and the Gaeltacht also recommended that provisional licence-holders take a full driving test within a specific period and that driving instruction should be given in school.
Penalty points must also be imposed on the last registered owners of illegally-dumped cars.
The committee met Mr Varadkar in July as part of a new process which enables members to consider legislation at an early stage and have a more meaningful input into law making.
Chairman Ciarán Lynch said the committee broadly welcomed the provisions of the new bill but made a few suggestions to make the law more effective.
“On a practical level, we believe that by discounting fines for parking tickets motorists will be encouraged to pay in timely fashion, such a system has proved very effective in the north,” said Mr Lynch.
In the North, the amount of a parking ticket is cut from £60 to £30 if paid within two weeks, but is increased after 28 days.
Mr Lynch added: “We are also recommending that a more stringent system be introduced to encourage more responsible disposal of cars to help combat illegal dumping, which is a serious problem in some parts of the country.”
The committee said advice is needed from the Attorney General on the constitutionality of taking blood specimens from unconscious drivers. It also raised concerns over proposals to move the tests for roadworthiness of commercial vehicles to a centralised location and away from local garages, fearing it could impact on employment.
Members asked for clarification on the implications of a planned centralisation of the driving licence system away from local authorities.