First of all, after Hugh's detailed explanation and alexis' link I realized that I was saying erroneously "upward expansion" while I meant "upward compression"! Sorry for that!
No problem, but still very different things.
'Upward compression' means compression, followed by a gain stage (ie. make-up gain). The compressor pulls the loud stuff down but leaves the quiet bits alone. The gain stage pushes the overall level up so that the quiet bits are made louder and the louder bits end up more or less where they were... so the low levels are raised.
However, the dynamics of the loud bits are still being mangled by the attack and release time constants of the compressor -- and that doesn't happen in the parallel compression technique because the loud bits aren't being processed at all (the very heavily compressed path signal is being completely drowned out by the clean direct path signal).
Consequently, upwards compression and parallel compression, while having a similar effect on the low level content (clean gain only), sound completely different on the high level content (compressed versus uncompressed) -- and that's why the two techniques and are used for different things.
Does this apply only when the mix of compressed and uncompressed signal is at 50-50%?
Technically, yes, but that is the optimum ratio. More direct signal weakens the low level lift effect, and more compressed signal sounds horridly over-compressed!
I apply some make-up gain in order to make the compressed signal to be as loud as the uncompressed one, wouldn't that make the transients louder?
Yes, of course -- but that's just the effect of adding gain to the processed signal. The action of the compressor is what is important when comparing compression techniques. An upwards expander can be arranged to make transients louder without affecting lower level signals and that would sound very different to a compressor followed by some make-up gain.
Is there any other way to achieve louder transients using a compressor?
Not in any practical sense, no.
Technical Editor, Sound On Sound