Time-series data is prevalent in many applications like smart homes, smart grids, and healthcare. And it is now increasingly common to store and query time-series data in the cloud. Despite the benefits, data privacy concerns in such outsourced services are pressing, making it imperative to embed privacy assurance mechanisms from the outset. Most existing related works have been focused on querying for different types of aggregate statistics. In this article, we instead focus on the secure support for advanced interval skyline queries, which allow to identify time series that are not dominated by any other time series within a query time interval. This is valuable for time-series data analytics in applications like remote health monitoring (e.g., identifying patients with high heart rates in a certain week). We present OblivTime, a new system framework for oblivious and efficient interval skyline query processing over encrypted time-series data. OblivTime is built from a synergy of time-series data analytics, lightweight cryptography, and GPU parallel computing, achieving stronger security guarantees and lower online query latency over the state-of-the-art prior work. Extensive experiments demonstrate that OblivTime can achieve up to$666\times$speedup in online query latency over the state-of-the-art prior work.