This is true.
Out of curiosity, I looked up all known safari parks within the United States, and there are... so many. Most are unaccredited, roadside facilities with questionable animal care and a severe lack of oversight. It is quite staggering just how many people will pimp out exotic animals to make a buck.

I found nine parks that were AZA-affiliated, with proper, navigable websites, and decent reviews. Additionally, there were two in Canada and one in Mexico. The U.S. ones are as follows:
-African Safari Wildlife Park (Port Clinton, OH)
-Fossil Rim Wildlife Center (Glen Rose, TX)
-Lee G. Simmons Conservation Park & Wildlife Safari (Ashland, NE)
-Lion Country Safari (Loxahatchee, FL)
-Northwest Trek Wildlife Park (Eatonville, WA)
-San Diego Zoo Safari Park (Escondido, CA)
-Safari West (Sonoma, CA)
-Wildlife Safari (Winston, OR)
-The Wilds (Cumberland, OH)
Note that this list does not include conservation centers, most of which are not open to the public (the largest of which is the Smithsonian Conservation Biology Institute) and which do not offer a "safari" experience in terms of amenities like camping, kayaking, and the like.
Four of the above safari parks are either directly controlled by, or have a formal affiliation with, an AZA-accredited zoo. Omaha's Henry Doorly Zoo manages the Simmons Conservation Park, Point Defiance Zoo & Aquarium manages Northwest Trek, Columbus Zoo and Aquarium has a formal affiliation with, but does own or manage, The Wilds, and of course San Diego Zoo and San Diego Zoo Safari Park are jointly owned/operated.