As mentioned, a baby should never travel in the car by anything but a child seat that is properly installed (preferably rear-facing). Do not take them out of the seat for any reason while driving: Not to feed, not to console them, not for anything but extreme situations. Always pull over first or find a parking spot. Accidents only require a fraction of a second.
Next to that the NHS advises that The Royal Society for the Prevention of Accidents recommends taking at least a 15-minute break every two hours [1]. Multiple other sources support this "two-hour rule", stating that prolonged travel may lead to breathing and spine issues due to being in an upright position. [2], [3], [4]
Research seems to agree that the less time spent in a car seat for a child that age the better (some sources going so far as to claim that two hours a day should be a maximum), it is understandable that this is not always an option. It is recommended to keep the travel time to an absolute minimum where possible.
Another option would be for you to invest in a lie flat car seat [5] (though their safety is disputed, see the comments). Make sure it confirms to all child safety guidelines. These would allow for longer travel per day (The two-hour rule still applies) because of a more comfortable positioning. Or, ideally, persuade your parents to come to you!
Make sure that whatever you decide is in the best interest of both the baby and its mother. An infant is a fragile thing, its safety and comfort is more important than that of any other parties involved.