Arguing about whose parents were here first is a bit of a distraction....
Dragging the topic back to "illegal immigrants" ....
More often than not by drug dealing and human trafficking....
Certainly not paying taxes.
Actually, every "illegal immigrant" pays taxes. Consumption taxes in the UK (VAT etc) are some of the highest in the world. Everyone pays them, even "illegal immigrants". "Existence taxes" like council tax are also endemic in the UK. And pretty high. As with VAT council tax doesn't ask for your PAYE number and therefore everyone is equal. There are various other taxes that the "illegal immigrants" pay at the same rate as everyone else - from VED to TV tax.
What they don't pay is only that tax which is imposed directly on earnings. We are talking PAYE (Income Tax) and NI (National Insurance)*. But, it's not always true. There are a lot of illegal immigrants who have National Insurance numbers and they end up paying these taxes too.
*Excluding CGT, Inheritance tax etc. for simplification and because they aren't usually applicable to II
My points: 
1. The term "illegal immigrant" does not mean someone who doesn't pay 
any taxes.
2. It does not even mean someone who does not pay IT and NI.
3. It does mean someone who doesn't qualify for the benefit/pension umbrella he is contributing to.