Nothing makes a bad first impression like underdressing for an event. It's the first thing the parents will notice about you, and no matter how politely you act, you won't be able to distract them from an egregious fashion faux pas.
Show respect by dressing appropriately. What's appropriate, of course, depends on the occasion. But even if it's a low-key family picnic, this is no excuse to come with tattered jeans and a wrinkled T-shirt. Instead, opt for clean, ironed khakis and a polo shirt.
For dinner at the parents' house, you can't go wrong with dark pants and a button-down shirt. If it's a nice dinner at a fancy restaurant, make sure to add a coat and tie to your ensemble. The restaurant may even have a dress requirement for this, and it would be more than a little embarrassing to have to don the restaurant's oversized spare coat. And if it's a wedding or similarly formal occasion you've been invited to, dress to the nines with your best suit.
What's appropriate also depends on parents' expectations and cultural traditions. It would be a good idea to have the girlfriend approve of your wardrobe choices beforehand.