The paper aims to compute approximate solutions of nonlinear algebraic systems with \(n\) unknowns and \(n-1\) equations, whose solution manifolds are curves. A continuation algorithm with certified existence and connectivity is proposed. It is abbreviated as ParCont and involves interval analysis and parallelotopes. The authors show that using parallelotopes instead of boxes brings many advantages, such as a better fitting to the shape of the manifold. Numerical features of the algorithm are addressed as well as a comparison with other methods. Some examples are provided, including a robotics application.