first things first, a chain of unconfirmed transactions != double spending.

What you probably have read is that when your incoming transaction uses the outputs of an unconfirmed parent transaction, there is a risk the creator of the parent transaction double spends the input of the parent transaction, and every other transaction in the chain gets cancelled. This is true.
If you trust the person who is using unconfirmed unspent outputs to build new transactions, this does not HAVE to be a problem. However, it is true that your transaction can not be confirmed BEFORE all parent transactions of your transaction are confirmed (they can all be put in the same block tough, just not in an earlyer block).