Palpatine and Vader have said they felt a disturbance in the Force when Luke started using it and there has been other times when Sith Lords said they felt a disturbance, but do they also feel a disturbance when a lot of people die like the Jedi do? Do they just not mention it because they usually caused it or they "like" it?
I can find no mention of a Sith sensing a disturbance in the Force due specifically to mass death, and Wookieepedia's article on disturbances in the Force doesn't mention anything, either.
That said, the Sith can definitely sense disturbances in the Force. As mentioned in the question, the Sith can sense a disturbance in the Force due to the awakening of a powerful user of the light side of the Force. They can also sense that someone has been harmed:
I sense Lord Vader is in danger.
Darth Sidious, Star Wars Episode III Revenge of the Sith
In the canon novel Lords of the Sith both Vader and Sidious sense a disturbance in the Force due to a threat to the lives of many Imperials (including themselves) on board the Star Destroyer Perilous:
[Vader and Sidious] surveyed the bridge in a silence broken only by the rasping of Vader’s respirator while the Perilous crossed parsecs in a blink. After a time, the bridge crew broke into a different rhythm as they prepared to return the Star Destroyer to normal space.
“Coming out of hyperspace,” called the helm.
“Coming out of hyperspace. Aye,” the call echoed up the bridge.
“And the test begins,” the Emperor said.
Vader looked at his Master, head tilted in a question, not taking his meaning until he, too, felt the disturbance in the Force.
As the Perilous emerged from hyperspace on the outer edge of Ryloth’s system, the mammoth viewport undimmed, giving a view of several distant gas giants and the nearby belt of asteroids that divided the outer system from the inner. The system’s star burned orange and bright in the distance. Ryloth itself was too far away to be visible.
“Maximum acceler—” began the captain, but before he finished an impact sounded from starboard and the huge starship vibrated.
Heads came up from stations and looked questions at one another. A second impact followed hard after, then a third, larger than the rest, caused the ship to list. Vader eyed the viewscreen, saw nothing. His Master stared at the floor, a strange half smile on his face.
“Situation!” the captain ordered, his voice calm.
Another impact shook the ship, a fourth, another, another. The ship listed farther. Alarms blared.
“We have electrical shorts and a few fires all over the ship,” the duty officer called.
In that case, Vader and Sidious sensed a disturbance in the Force due to mass injuries rather than death, and many of those injured Imperials would soon die when the Perilous was destroyed shortly after being attacked.
It stands to reason that the Sith can sense a disturbance in the Force due to mass death given that they can sense other, similar disturbances in the Force -- it's just not mentioned because the Sith usually cause such a disturbance and therefore don't need the Force to sense it. Sensing a disturbance in the Force has nothing to do with whether or not someone "likes" the event -- in many cases both the Jedi and Sith sense disturbances due to events which they don't like.