Connecting your phone to the battery pack doesn't directly connect the cells in parallel. I assume this is where your guess of an equilibrium with equal voltage -> equal charge percentage comes from.
Shorting lithium-ion / lithium-polymer (LiPo) cells together like that would likely cause one or both to literally catch fire from the high currents, or from overcharging / over-discharging. (Circuitry to prevent over-discharging even if the charging cable is shorted out is absolutely essential).
There are some links to youtube videos of lithium battery fires on a recent electronics.SE question about designing your own charger. (TL:DR: it's way the hell too dangerous to consider doing for a homebrew design, because Lithium cells need a LOT of protection circuitry to be mostly safe.)
So the whole idea of connecting cells together and letting their voltages equalize "naturally" is just completely not viable for modern batteries.
Chargers use DC-DC switching power supplies to charge at constant current. They use inductors to efficiently convert to a different voltage (higher or lower). (For example, to produce a lower voltage, see this detailed explanation of a buck converter doesn't use a transformer, just an inductor. Also a discussion of multi-phase buck converters used on computer motherboards.)
In the water analogy, where water represents charge and pressure represents voltage: A converter is like a pump that can move charge from a lower reservoir to a higher reservoir. (voltage = pressure = gravitational potential energy (per unit volume / charge).) A small fraction of the energy transferred is lost to inefficiencies in the conversion. (Maybe a couple %, IDK).
Since the capacity of the external battery pack is larger than the capacity of the phone's battery pack, it should be obvious that moving charge from the large reservoir to the small reservoir can take the phone battery from 5% to 100% while only dropping the battery pack from 35% to 12%. I don't think this is what the question was really about.
Just to make it even more obvious why batteries aren't just connected together to equalize: Some batteries may have multiple cells in series instead of one large cell. This is typically for physical design reasons, more than to get a higher voltage, because DC-DC converters will be used anyway to produce supply voltages in the 1 V to 2 V range to power most electronics.
Since Lithium cells are so finicky and dangerous, wiring them in parallel instead of series is unwise. One cell could end up taking most of the current. So instead, they're wired in series with circuitry for each cell to bypass it before it overcharges or undercharges.
The power transfer between phone and battery pack happens over a USB cable, which runs at 5 V. (Or, with USB power delivery signalling, the device being charged can signal that it can accept up to 20V, allowing for higher power at the same current to reduce resistive losses and allow faster charging without exceeding safe current limits for the cable / connectors.)