Parametric model checking (PMC) computes algebraic formulae that express key non-functional properties of a system (reliability, performance, etc.) as rational functions of the system and environment parameters. In software engineering, PMC formulae can be used during design, e.g., to analyse the sensitivity of different system architectures to parametric variability, or to find optimal system configurations. They can also be used at runtime, e.g., to check if non-functional requirements are still satisfied after environmental changes, or to select new configurations after such changes. However, current PMC techniques do not scale well to systems with complex behaviour and more than a few parameters. Our paper introduces a fast PMC (fPMC) approach that overcomes this limitation, extending the applicability of PMC to a broader class of systems than previously possible. To this end, fPMC partitions the Markov models that PMC operates with into \emph{fragments} whose reachability properties are analysed independently, and obtains PMC reachability formulae by combining the results of these fragment analyses. To demonstrate the effectiveness of fPMC, we show how our fPMC tool can analyse three systems (taken from the research literature, and belonging to different application domains) with which current PMC techniques and tools struggle.