Do You Need to Install a Ground Rod for a sub panel in a detached structure?

Have a licensed electrican advise you in person.

Yes, and you also need a means of disconnect either outside or at the point where the sub feed enters the building.

NEC 250.32 (D) (2)

(D) Disconnecting Means Located in Separate Building or Structure on the Same Premises. Where one or more disconnecting means supply one or more additional buildings or structures under single management, and where these disconnecting means are located remote from those buildings or structures in accordance with the provisions of 225.32, Exception No. 1 and No. 2, 700.12(B)(6), 701.12(B)(5), or 702.12, all of the following conditions shall be met:

(1) 

The connection of the grounded conductor to the grounding electrode, to normally non–current-carrying metal parts of equipment, or to the equipment grounding conductor at a separate building or structure shall not be made.

(2) 

An equipment grounding conductor for grounding and bonding any normally non–current-carrying metal parts of equipment, interior metal piping systems, and building or structural metal frames is run with the circuit conductors to a separate building or structure and connected to existing grounding electrode(s) required in Part III of this article, or, where there are no existing electrodes, the grounding electrode(s) required in Part III of this article shall be installed where a separate building or structure is supplied by more than one branch circuit.

(3) 

The connection between the equipment grounding conductor and the grounding electrode at a separate building or structure shall be made in a junction box, panelboard, or similar enclosure located immediately inside or outside the separate building or structure.

if your running a sub main run an appropriate sized  earth from the board the sub-main originates from. Local installation rules will cover that depending where you are.

If there is already a ground at the main panel, you can ground the subpanel to that.  (In fact, you really only want to have one point of grounding, anyway, or else it's not really grounded.)