Add items to a list, or update the quantity if already in the list
I am not really sure about the needs but when you see something like this it should make you question if there is a better data structure.  If you stored the parts in an Object for example keyed by partNumber the code is more algorithmically efficient (O(1) vs O(N)) and a bit cleaner.

var newPartNumber = partToAdd.partNumber,
    existingPart = parts[newPartNumber];

if(!existingPart) {
    parts[newPartNumber] = partToAdd;
}
else {
    existingPart.quantity = ...;
}


Now if there are other reasons for using an array or using an object keyed by part number wouldn't be useful anywhere else I would use the some
method.  

 var partExistsOnTransfer = transferList[0].parts.some(function (part, indx) {
    if (part.partNumber === partToAdd.partNumber) {
        // if part already exists, add to existing quantity, do not push new part onto list
        part.quantity = (Number(part.quantity) + Number(partToAdd.quantity)).toString();
        return true;
    }
});


Also remember returning false on an Array forEach doesn't make it stop iterating.