Yes, her father was a well-to-do business man. Not old money and not insanely wealthy, but rich enough to send her off to finishing school in Switzerland.
Unfortunately, he turned out to be a real jerk -- he ran off with most of the money she earned from her first few films. At his advice she put the money in trust with him as the trustee, and then he absconded with it. It was a huge blow to her; not so much the money but because she'd absolutely idolized her father.
(I got all this from the Gene A&E bio on the Laura DVD.)