Based on the special power series with Hadamard gaps, a family of hyperbolic complete minimal surfaces located between two parallel planes in $ {\mathbb{R}^3} $ was studied. Firstly, the following results were obtained: Let $h(z) = \displaystyle \sum\limits_{j = 1}^\infty {{a_j}{z^{{n_j}}}}$ be a series with Hadamard gaps, where $ z \in \mathbb{C} $, $j = 1,2, \cdots $, and satisfy three given special conditions. Then for all divergent paths $ \gamma $ in the unit disk $ \Delta $, $\displaystyle \int_\gamma {{{\left| {h'(z)} \right|}^2}\left| {dz} \right|} = \infty$ satisfies. At the same time, the specific analytical functions satisfying the above conditions were listed. Secondly, by selecting the appropriate Weierstrass representation pair and using the above conclusion, the hyperbolic complete minimal curved surface family and its specific form between two parallel planes in $ {\mathbb{R}^3} $ were constructed.