If one were to add the compressed signal back to the original in this process in order to raise lower levels up we have a form of parallel compression again.
Again, unless I'm also not making a connection, the difference between parallel compression and simple upward compression is what happens above the threshold setting. In parallel compression, the answer is "a wet/dry mix of downwards compressed and uncompressed versions of the signal." Whereas in simple upwards compression, the answer is, "nothing".
Let's say, just for illustration, that the OP's problematic piano notes are two notes at -13 and -16 dBFS. With upwards compression, one could set the threshold to -10dBFS and a compression ratio of 3:1, just for example, and those two notes would come out the other end at respective amplitudes of -11 and -12dBFS, and there would be zero change to the amplitudes or energy distribution of any if the signal exceeding the -10dBFS threshold.
One could synthesize part of that with downward compression followed by makeup gain as you say. Set the threshold to -19dBFS, downward compress by 3:1, and then apply a linear gain to the result of 6dB. But you'd also be affecting everything else in the signal above -19, all the way up to the peaks, and not just the stuff below -10.
Granted, when you go parallel, you wind up with a compression curve that tends to boost the lower level stuff and tends to leave the higher stuff alone, so the peaks themselves will not be changed much, if at all. But that curve has a very soft knee. That's fine if you're looking to boost the overall juice of a track. And I'm not saying there's anything wrong with that.
But it *is* different from simple upward correction which (unless you design a custom curve, which can be done with a dynamics processor) has a sharp cutoff at the threshold. It is a much more surgical approach when you are looking to target a specific slice of the dynamic range, like just a couple of errant notes or drum hits.
Additionally, though this admittedly goes a little beyond the OP, the tools that offer upward compression offer even more flexibility than just that. With Neodynum, you can set up to four different dynamics zones with virtually any combo of upward or downward compression or expansion. Additionally they can be applied to a user-defined custom EQ "key" curve, adding to the surgicality of the process. Similarly, with a dynamics processor, you can draw a custom expansion/compression curve to fit the track's needs like a glove.
So, all in all, yes, there is a difference in effect. But more than that, there's a difference in definition. Parallel compression employs downward compression to do it's thing, even if it's thing winds up being similar, if not exact, to what upward compression actually does.
G.