Genuine TTL logic has the property that a floating input is treated as a logic 1 inside the chip. Thus for standard 54/74xx, 74ls, 74ls, 74f and 74als if you leave a gate open it will be seen as high. It is better to pull up to a defined level noise wise, but it will work. This is because the inputs are the emitters of a multi emitter transistor, and the inputs that are pulled down are active in determining logic state.
Parallel connection of the outputs of gates in the same package if they have the same inputs is doable, it just increases the current sink capacity and slightly increases source capacity, and makes the edges faster into high capacitance loads. Not a good design recommendation but it works.
Replace the IC with a plain 74xx device, or a 74ls or 74s one. This is one of the edge cases where the cmos parts will not work at all. The dual gates in this case are to drive the long cabling of a test lead with fast edges. If you can put pull ups to all those unconnected pins, or tie them together to the input ( in this case it will not increase the fan in loading at all) to improve things. Then you can use the 74 HC or HCT parts as well.