We introduce a new parametric kernel function, which is a combination of the classic kernel function and a trigonometric barrier term, and present various properties of this new kernel function. A class of large- and small-update primal-dual interior-point methods for linear optimization based on this parametric kernel function is proposed. By utilizing the feature of the parametric kernel function, we derive the iteration bounds for large-update methods,O(n2/3log⁡(n/ε)), and small-update methods,O(nlog⁡(n/ε)). These results match the currently best known iteration bounds for large- and small-update methods based on the trigonometric kernel functions.