According to the Wikia's Animal Crossing - Flower page, there are 3 varieties of carnations, Pink carnations can only be obtained on Mother's Day (May 10th). Red carnations can only be obtained on Father's Day (June 21). White carnations can be grown by placing a red carnation and a pink one to create a hybrid.
You can also grow red and pink carnations, but only from existing red and pink carnations. In other words, you cannot grow any pink carnations without at least 2 pink carnations. The same thing is true of red ones. With only red carnations, you will never be able to grow pink ones, and vice versa.
Your best bet would be to try and find a person who actively grows carnations and borrow some from that person's town. Otherwise, you will have to wait until May 10th before you can get pink carnations.