EDIT: I crossposted with @Grubermensch. This answer is really similar to his one but expands a little bit more on homebrewing in the end, so I'm leaving it here anyway.
I know one system that might qualify, depending on what you mean by separately leveling up.
If you want to get two separate XP tracks, one for race and one for class, you might as well ignore my answer.
If you want a race that improves along class features, as if it was a parallel class such as KRyan's gestalt suggestion, Legend can do that.
Let's start by saying that Legend is a heavy hack of D&D 3.5e.
As heavy that it's completely unrecognizable in some aspects and one of those is character classes.
First of all I need to explain you the basics of character creation:
- Tracks are sets of seven related class features (called the seven circles of that track), unlocking as the level goes up;
- Every class is made up of one chassis (saves progression, attack bonus, hit die) and three tracks. This allows for what I call vertical multiclassing: you don't stop progressing as a paladin to start again with monk features, instead you lose some class features from paladin and gain some monk ones, swapping out your whole track and getting seven synergistic circles;
- Except for the first level where two first circles are gained, circles are unlocked one per level, drawing from the three different tracks in alternance - this means that every track has one circle unlocked every three levels (again, except for the track unlocking the second circle at level 3, because it also unlocked the first circle at level 1).
While most races are just a set of ability score and skill modifiers, some races such as Dragon or Vampire are full fledged tracks (and a chassis that has to be used instead of the class one).
This means that, if you chose one of those races, they level up along your character and are as important and character-defining as your other tracks.
A Dragon gets seven dragon-related racial features (the dragon track's circles) that unlock one after another, every three levels.
While the manual has sample races, drawing from the classic D&D Player Handbook's ones, there are rules for creating new "regular" races and creating a new racial track is on par with homebrewing a new track (which is routinely done and balanced against the other tracks on their forums).
This is to say that creating a new track+chassis for a race such as human does not really go against the existing races (which are just examples) and making a racial track mandatory is not unbalancin the system (at most it's limiting the possible character builds).