Software Fault Injection (SFI) is a widely used technique to experimentally assess the dependability of software systems. To provide a comprehensive view on the dependability of a software under test, SFI typically requires large numbers of experiments, which leads to long test latencies. In order to reduce the overall test duration for SFI, we propose FASTFI, which (1) avoids redundant executions of common path prefixes for faults in the same injection location, (2) avoids test executions for faults that do not get activated, and (3) utilizes parallel processors by executing SFI tests concurrently. FASTFI takes patch files that specify source code mutations as an input, conducts an automated source code analysis to identify the function they target, and then automatically parallelizes the execution of all mutants that target the same function. Our evaluation of FASTFI on four PARSEC benchmarks shows a SFI test latency reduction of up to a factor of 26.