As you concluded, alchemy does require energy for a transmutation to take place.
You mention that the energy comes from the departed souls travelling through the gates in the parallel universe, however that parallel universe only exists inside the 2003 continuity. There is no such parallel world in canon. While this explanation for the 2003 anime makes sense, it also leaves questions since we do see failed transmutations leaving alchemists completely exhausted. Being that the 2003 anime isn't based on canon, this is probably just an oversight.
The FMA Wikia on Alchemy has this to say about the subject:
In the 2003 anime, though it is never outright explained from where the energy in transmutation comes, alchemists who attempt to create objects outside their range of skill are left exhausted, suggesting that at least some of the energy comes directly from their bodies. At the end of the anime series, it is explained that the energy powering transmutation is actually the energy of departed human souls passing through The Gate to the alchemical world from our world, negating the concept of Equivalent Exchange altogether.
In Full Metal Alchemist: Brotherhood, this energy comes from different places depending on if you are using alchemy or alkahestry.
Normally for transmutation to take place, a transmutation circle must be drawn. The purpose of a transmutation circle is to channel the energy needed for the transmutation. This energy is channeled from energy that already exists in nature.
The FMA Wikia has this and more to say about transmutation circles:
The circle itself is a conduit which focuses and dictates the flow of power, tapping into the energies that already exist within the earth and matter. It represents the cyclical flow of the world's energies and phenomena and turns that power to manipulable ends.
During the story we are told the energy for transmutation in Amestris is powered by tectonic shifts in the earth's crust.
Later, it is revealed that all transmutations use energy from a giant underground network of philosopher's stones, which go through Father's body.
However, the transmutations that take place outside in Xing use the Dragon's Pulse as an energy source.
To quote from the Wikia's Alkehestry page:
Whereas Amestrian Alchemy claims to have its roots in the energy of tectonic shifts and practices manipulation of matter toward scientifically practical ends, Alkahestry is centered on a concept called the "Dragon's Pulse" which speaks of the Earth itself having a constant flow of chi (life energy) which flows metaphorically from the tops of mountains down to the land, nourishing everything it passes with that energy as does blood coursing through the veins.