@Mittzy is absolutely correct, you do need a relay. The factory horn draws about 3 amps and the factory switch is just barely able to handle that.
You could trigger the relay and the OEM horn at the same time. The OEM horn switch is switching the negative not the positive wire. So keep that in mind when installing the relay.
Definitely use a relay. I learned the hard way when I installed air horns. Ended up replacing the horn button. Went with the Eastern Beaver relay setup and no problem ever since.
There is a very old but simple rule. Whenever you have doubts - there is NO doubt.
USE A RELAY.
The Triumph one is anecdotal to pass laws. Nothing more.
As long as you fit (in the EU) another € approved horn - there is no reason at all to keep the original.
IF there was adequate current capability, it would be parallel to answer your basic question.
However that horn draws 9A - https://fiammcomponents.com/media/369586/7211223.pdf
If you were connecting in parallel with the OEM horn, you would be sending about 12-14A through the horn switch (which would the element of greatest concern)
Definitely requires a relay
If you do plan to install a relay for this horn anyway, it just makes more sense to wire both the OEM and new horns supplied (in parallel) from the relay, with the OEM horn wires driving only the relay
Keeping the stock horn back in 2005 when I bought my first Rocket I thought that something was wrong with the horn i replaced it within the first week as I was to embarrassed to use it.