If you wish to proceed with a PICAXE 08M and a basic parallel LCD as per your link you will need some form of interface.
Rev Ed sell the FRM010 chip for UK£4-00 (+postage) which can handle the requirements for you.
See here:
http://www.picaxe.com/docs/frm010.pdf
and:
http://www.techsupplies.co.uk/epages/Store.sf/en_GB/?ObjectPath=/Shops/Store.TechSupplies/Products/FRM010
If you change to a 28X1 then you will have 8 pins predefined as outputs and can re-allocate some port C pins as well as outputs if your project needs more outputs.
With any PICAXE chip having sufficient pins, that is at least 6 outputs no interface chips is needed for a parallel connection.
The difference:
If you use an LCD with a serial input, not only do you only need just 1 PICAXE output, but you can use the inbuilt SEROUT command for the serial comms to the LCD.
If you use a parallel interfaced LCD you need to add your own subroutine to pass the characters (as data) to the LCD display. There are plenty of example subroutines posted on this forum to copy from.
Keep in mind that a few folks do seem to come to grief initially trying to get parallel interfaces working.
That Hobby-2-go and many E-Bay LCD offerings do not indicate which controller protocol it uses. Most use the Hitachi HD44780 protocol for commands but not all.