I am reading Apostle's Modular Functions and Dirichlet Series in Number Theory. Theorem 1.7 says that a non-constant elliptic function has at least 2 simple poles or a double pole in each period parallelogram. And the book went on saying that these are the only 2 possibilities, and that they led to two different theories proposed by Weierstrass and Jacobi.
I am wondering why cannot we have a triple pole or just any poles of orders higher than 1.
You can have poles of higher orders. For example, $\wp^2$ has poles of order $4$ in every point of the period lattice. Also the derivative $\wp'$ has a pole of order $3$.
A theorem of Abel says that you can find an elliptic function with any prescribed zeros and poles $\alpha_i$ and orders $m_i$ (positive $m_i$ being the order of a zero and negative $m_i$ the order of a pole) if and only if $\sum m_i = 0$ and $\sum m_i \alpha_i \in L$ lies in the lattice.
This excludes the possibility of a single simple pole $\alpha$ already: the first sum forces the existence of exactly one simple zero, and the second equation forces that zero to appear at $\alpha$.