We've heard about stories of people dying from carbon monoxide inhalation inside a parked car inside a garage. So will backing into a garage when you park cause similar health issues or even death?
Ultimately if you park a car inside a garage that is under your home, then there will always be a chance and small percent of fumes that do linger in the garage. Some of those fumes could travel into your living area. The insulation, walls, windows, doors and air flow play a big roll on whether this will affect your health. The better the air flow, then the less amount of fumes will be trapped in your home.
If you back into your garage, versus head in, your tailpipe will be pointing into the home, versus towards the garage. Some people have set timers on an electric fan to suck up some of those fumes each time the start up their car. This will help get rid of some of those fumes.
So depending on your comfort level, concern for health, ease of access and preferences, it should still be safe to back into your garage. But ultimately look at your garage, see if there's proper ventilation and you decide what you want to do.