Binary response data arise naturally in applications. In general, the well-known logistic and probit regression models form the basis for analyzing binary data in practice. These regression models make use of symmetric link functions (logit and probit links). However, many authors have emphasized the need of asymmetric links in modeling binary response data. In this paper, we consider a broad class of parametric link functions that contains as special cases both symmetric as well as asymmetric links. Furthermore, this class of links is quite flexible and simple, and may be an interesting alternative to the usual regression models for binary data. We consider a frequentist approach to perform inferences, and the maximum likelihood method is employed to estimate the model parameters. We also propose residuals for the link models to assess departures from model assumptions as well as to detect outlying observations. Additionally, the local influence method is discussed, and the normal curvatures for studying local influence are derived under two specific perturbation schemes. Finally, an application to the coca leaf cultivation in Peru is considered to show the usefulness of the proposed link models in practice.