In case the other answer is a little to direct...
Think of the start switches and the stop switches as two separate items with separate wiring. Either of the stop switches must interrupt the current to the motor starter coil momentarily, so the wire must go from the source, through one stop switch, which is normally closed and then from that switch to the other switch, through it to the starter. (I.E. series) During motor run, the current to keep the starter coil in uses this route also going through an auxilliary contact and the overload contact on the starter and returns to the source (panel, control transformer, etc.) along the common. So any one of the contacts, overload, auxilliary, or stop opens, the motor starter will open and the motor will stop.
Either Start switch must provide momentary current to the starter coil to pull it in allowing the above contacts to all close, maintaining the power to the coil. In order to do this, the current must be available at one side of both start switches which are normally open, at all times. (parallel) When either button is pushed the current then flows to the coil pulling it in.
So to repeat the simple answer. Start contacts in parallel, stop contacts in series.