Yes, not only do cyclists personally save money on transport costs, parking etc, they save the country money too !
A study in Australia release by deputy prime minister last year found that cyclists in Sydney save the economy AU$21 (NZ$23) for every 20 minutes trip to work and back and AU$8.50 (NZ$9.35) for each pedestrian doing a similar trip. "The economic benefits of riding and walking to work include better health, less congestion, reduced infrastructure costs, reduced greenhouse gas emissions, better air quality, noise reduction and savings in parking costs." (http://www.smh.com.au/national/bike-riders-save-economy-21-on-each-commute-20130730-2qxdg.html).
Closer to home, research by the University of Auckland’s School of Population Health found that "... the best kinds of policies involve investing in specific road changes that effectively improve cycling safety while also helping a wide range of people feel safe while riding." and also that "By far the largest benefits come from reducing deaths related to lack of exercise – a spend of $600 million on the right kind of cycling infrastructure yields savings from increased exercise in the tens of billions of dollars." (https://www.auckland.ac.nz/en/about/news-events-and-notices/news/news-2014/02/cycling-infrastructure-more-economic-than-for-other-transport.html).