There are at least two 123reg hosts. You're using the one in the UK. They started life as a registration company and offer simple hosting suitable for small static sites only. Last time I looked their hosting didn't include PHP or MySQL.
This looks to me like a strategy of avoiding that part of the hosting market that requires the heaviest investment in support and service (no laughing in the back row please, there are web hosts who do provide these, you just need to
choose carefully). So you can upgrade to a hosted e-commerce cart, or to a dedicated server through a sister company (where most likely you'd find yourself responsible for managing it yourself).
I suspect that they are also trying to maintain differentiation with services offered by other parts of their ultimate parent, e.g. via Tiscali for consumers and Pipex increasingly focused on businesses.
So, digression over, sorry, unless something has changed, the problem is that you hosting account can't run dynamic applications like Zen Cart.