Reduced-order models are indispensable for multi-query or real-time problems. However, there are still many challenges to constructing efficient ROMs for time-dependent parametrized problems. Using a linear reduced space is inefficient for time-dependent nonlinear problems, especially for transport-dominated problems. The non-linearity usually needs to be addressed by hyper-reduction techniques, such as DEIM, but it is intrusive and relies on the assumption of affine dependence of parameters. This paper proposes and studies a non-intrusive reduced -order modeling approach for time-dependent parametrized problems. It is purely data-driven and naturally split into offline and online stages. During the offline stage, a convolutional autoencoder, consisting of an encoder and a decoder, is trained to perform dimensionality reduction. The encoder compresses the full-order solution snapshots to a nonlinear manifold or a low-dimensional reduced/latent space. The decoder allows the recovery of the full-order solution from the latent space. To deal with the time-dependent problems, a high-order dynamic mode decomposition (HODMD) is utilized to model the trajectories in the latent space for each parameter. During the online stage, the HODMD models are first utilized to obtain the latent variables at a new time, then interpolation techniques are adopted to recover the latent variables at a new parameter value, and the full-order solution is recovered by the decoder. Some numerical tests are conducted to show that the approach can be used to predict the unseen full-order solution at new times and parameter values fast and accurately, including transport-dominated problems.