You need the part that connects the two cables together. It is a small rectangular piece with raised ends, making for a "U" shape. There is a hole in the center with two slots extending outward from the hole. The little ball on each cable end goes into the hole in this connector piece and the cable itself slides outward to each end. The cables meet in the center, under the transmission and the tension on them keeps them in place.
Attaching just the driver's side obviously won't work because it's length is too long, but when connected to the other side's cable it shouldn't be. The connector is about one inch in length. There are different lengths of this connector for the different model cars. Your cables are the length for the one inch piece and if you install a longer one, you might not be able to adjust the brake properly.
The adjuster bracket (Equalizer) with the long threaded bolt is designed to slide along the length of the driver's side cable. The connector is available from any Mustang parts dealer. National Parts Depot has them listed as: Connector, Park Brake Cable for 67-73 Mustang P/N 2A790-1A Price is $6.50.
Be aware that the lengths of the cables are different from one year to another and 1967 was no exception. In 1967 the length of the right and left side cable is the same and can be swapped from side to side. This is not true for the 66 or the 68.
It's anybody's guess as to why they changed the design every year. It's just another quirk in the Mustangs evolution. | 
'67 289 coupe parking brake issues 
for anyone else interested in seeing the part I found it here 
