Can a mother who is a non-US citizen change name of minor without father's permission in Florida

Thnank you for your question. If you and the child are in the U.S.A. on medical parole from the Cayman Islands, you, the father and the child (who are all Cayman nationals) remain under the jurisdiction of the Caymans for purposes of your question. Under Cayman Law,  the following is from the applicable Cayman statute: "15. (1) Where a residence order is in force with respect to a child, no person may (a) cause the child to be known by a new surname; or (b) remove him permanently from the Islands, without either the written consent of every person who has parental responsibility for the child or the leave of the court." So, without the Cayman father's written consent or a Cayman court order, you may not change the child's surname under Cayman law.  Tha applcaible Cayman law is found at Children Law (2012 Revision), Part II Order With Respect to Children in Family Proceedings, Section 15, Change of Chjild's Name or Removal from Jurisdiction.  You may look at the statute for yourself at www.gov.ky/portal/pls/portal/docs/1/11525525.PDF.  Of course, as this is a queston of foreign law, you may wish to check with a Cayman attorney before reaching a final conclusion as to your course of action. Good luck!