No, you are entitled to all the legal rights and protections that are available to a full-timer. In addition, as a part-time worker (which includes someone on a zero hours contract) you have specific rights not to be treated less favourably than your full-time colleagues, under the Part-time workers (Prevention of Less Favourable Treatment) Regulations 2000.
Although you have rights not to be treated less favourably, remember that there is still an earnings test for SMP. You must have had average earnings of at least £116 a week in the eight weeks (or two months if monthly paid) before the 15th week before your due date. If you earn less than this, you may qualify instead for Maternity Allowance.