Due to the volume of applications, they cannot verify every assertion on the application. It should be noted that the punishment for including false information can be harsh including withdrawal of the offer of admission and expulsion.
Some schools do indeed use outside services as mentioned below.
HBS
"Verification: The School will verify application information and reserves the right to withdraw any offer of admission already made if there is any discrepancy between the self-reported information and information provided through verification."
UCLA FEMBA
"All applicants are required to acknowledge that he or she understands and agrees that any misrepresentation or omission of facts in an application is grounds for denial of admission, the cancellation of admission, or expulsion. By applying to the UCLA FEMBA program, applicants authorize the Admissions Committee to verify any information submitted.
The UCLA FEMBA program uses Kroll Background America to verify all admitted students who choose to enroll. All offers of admission are contingent upon successful verification of recommendations and employment information; applicants who choose to accept their conditional offer of admission must authorize the information verification and are charged an additional fee by Kroll Background America to conduct the information verification."