The most obvious example of symbolism in this story is the rocking horse itself. It is clear that is has a symbolic function as it obviously stands for both itself and for something beyond itself. There are many possible symbolic means that can be attributed to the rocking horse, and I will discuss two of them.
Firstly, it is clear that the rocking horse could be a symbol of supernatural forces. This refers to Paul's "secret within a secret" - the way that his frenzied riding gives him the name of the horse that will win the next race. It is important to examine how Paul is described when he is on the horse, and how this description increases in intensity as the story progresses. For example, repeated references are made to Paul's eyes:
"I've got to know for the Derby! I've got to know for the Derby!" the child reiterated, his big blue eyes blazing with a sort of madness.
And then finally, when he is discovered by his mother:
Then suddenly she switched on the light, and saw her son, in his green pyjamas, madly surging on the rocking horse...
"It's Malabar!" he screamed in a powerful, strange voice. "It's Malabar!"
His eyes blazed at her for one strange and senseless second, as he ceased urging his wooden horse.
Clearly words such as "powerful", "strange" and "blazed" suggest that what is happening to Paul is something supernatural.
The other major symbol of the rocking horse is that it is a symbol for Paul's desire for his mother's love. From the start we are told that the mother is unable to love her children - when she is with them, she feels the "centre of her heart go hard." Clearly aware of the need for money in the house from the voices, Paul has a conversation with his mother where he associates luck and money as being the same thing. Therefore it could be argued that Paul's adventures with the rocking horse are an attempt to gain his mother's love because everything else has failed.
You could also argue that the horse could be a symbol for greed or temptation, so have a look at the story again and see how you might justify these symbolic meanings.