Perhaps.  But I am not sure why you would want to change without the approval of the supported organization.

Supporting organization status is an IRS classification that allows many charities to be effectively considered public charities even though they do not meet the regular public support tests set out in sections 509(a)(1) or 509(a)(2) of the Tax Code.  (See Ready Reference Page: “Calculating Public Support Percentage”) The classification is open to those that are organized and operated exclusively for the benefit of the supported organization(s) and are “operated, supervised, or controlled by or in connection with such organization.”  (See Ready Reference Page: “Supporting Organizations Qualify as Public Charities”).

IRS Form 8940 allows an organization to request a change in its foundation status and its treatment for tax purposes, assuming, of course, that you currently meet the public support requirements.  But the change in status, by itself, would not change the supporting organization’s governing documents.  If you are organized exclusively to support the XYZ charity, you could be classified as a public charity but still have the limitation on your purpose to support the XYZ charity.  If the XYZ charity appoints all of the members of your board, the classification as publicly supported would not by itself change the way your board is appointed.  Well written bylaws would not let you change critical provisions of your governing documents like these without the approval of the supported organization.

You don’t say why you want to change your classification.  If you want to change only your classification, and not your powers or structure, you may be able to do it without approval, although there would be no reason not to obtain approval if you are going to continue your support.  If you are doing it to change your powers or your structure, you probably cannot change without approval.  But not all governing documents are well written to protect the interests of the supported organization, and you may be able to act unilaterally without any approval.  That’s not likely to engender good relations with your currently supported organization, however, and at least a “heads up” might help reduce the risks of litigation.