(A) Except as provided in Section 3-4-4 (Public Areas Where Restraint of a Dog is Not Required), an owner or handler of a dog shall keep the dog under restraint.
(B) A person holding a dog on a leash or lead shall keep the dog under control at all times.
Here is 3-4-4, essentially a list of all the legal off-leash spots in the city:
§ 3-4-4 PUBLIC AREAS WHERE RESTRAINT OF A DOG IS NOT REQUIRED.
An owner or handler may allow a dog to be without restraint as otherwise required by this title in the following places:
(1) Auditorium Shores from South First Street west to Bouldin Creek;
(2) the portion of Zilker Park bounded by Stratford Drive, Barton Springs Road, and Park River Road;
(3) the right-of-way of Far West Boulevard between Great Northern Boulevard and Shoal Creek Boulevard;
(4) the portion of Robert Mueller Municipal Airport land bounded by Old Manor Road, Manor Road, the airport fence, and Lovell Drive;
(5) Red Bud Isle east of Red Bud Trail;
(6) the portion of Onion Creek District Park south of Chunn Road;
(7) the portion of Northeast District Park bounded by Lake Long Road, Crystal Brook Drive, and the Missouri-Kansas-Texas Railroad right-of-way;
(8) the portion of Walnut Creek District Park bounded by Cedar Bend Drive, Walnut Creek, and the park fence on the west and east sides;
(9) the portion of Lake Austin Metropolitan Park bounded by Park Drive, the park fence on west side, Turkey Creek, and top ridge of bluff line that overlooks Lake Austin;
(10) Shoal Creek Hike and Bike Trail from 24th Street to 29th Street; and
(11) in an area designated by the Director of the Parks and Recreation Department.
Source: 1992 Code Section 3-3-3; Ord. 031009-9; Ord. 031211-11.