I have been flying A320s for over two years now and this is the first time I am hearing this. Normally the engines are always started using the APU with its bleed providing air for the air starter motor. So, in normal operations, cross bleed is set at AUTO, APU bleed is turned on and both the engines are started with the help of the APU. Once the engines are started APU bleed is turned off and the APU is shut down. Below is the after start flow of the A320. The green shows the flow of the pilot who starts the engines. As you can see, once the engine mode selector is set to norm, APU is shut down.
The procedure you are talking of is called a cross bleed start which is used when the APU is inoperative for the flight. It is also used by airlines that enforces one engine taxi to save fuel. When APU is not operational, we start one of the engines using an external air source. Once the engine is started the external unit is removed and then to start the next engine we use the bleed from the already started engine. To do that we open the cross bleed and increase the thrust on the running engine until the bleed pressure is sufficient enough for the second engine start up. As soon as both the engines are running we set the cross bleed to AUTO and revert back to normal operation.
As I have told before, I have never heard of any concerns regarding harmful gases coming of the APU bleed. Back when I used to fly Dash 8s we had issues in some of our airplanes smelling of burning oil once engine bleeds are turned on after start up. The airline consulted with Bombardier and they sent a bulletin which said that sometimes oil drippings from engine bearings can fall onto the bleed ducts. Because the duct temperature tends to be quite high, the oil burns off and the smell goes into the cabin. According to them, the lubrication oil used in airplanes is not hazardous or poisonous to human beings. I would say, it applies to all transport category airplanes with gas turbine engines because nearly all of them uses the same type of oil.