Helium balloons are pulled by gravity, as are all objects with mass. The reason they don't fall is that there is another force acting on them, a buoyant force from air pressure that is equal to the weight of the air displaced by the balloon.
The reason you don't float is that the weight of the air you displace is quite a bit less than your weight (a person is more dense than air). The reason a normal balloon doesn't float is that the weight of the air it displaces is just a little bit less than the weight of the balloon (because it is filled with air, but the rubber of the balloon itself is more dense than the air).
The analogy you want is to objects floating (or suspended) in water. Most rocks sink to the bottom, pulled by gravity, because the weight of the water they displace is less than their own weight. A bowling ball (ironically) is very close to the same density as water, so it will float suspended in mid-water, just like the helium balloon that has leaked a little bit.