I would never put any flash on any websites. Flash is an old tool to show animations. Just take in consideration that people are using more and more mobile devices like Iphones where flash cannot be visible. So why would you like to have flash on your website? You need to target all the possible audience for your website, regardless of what the user is using, mobile, desktop, laptop, etc. If somebody lands on your site, he/she lands there for a reason, probably because he/she was searching on Google some of your keywords. So you need to get that person interested in what you have on your website immediately, making a use of all that you have on your website, including whatever you want to show whith that flash. So why would you let's say "hide" some info from some people who are using iPhones for example, just because you have a flash which is not visible to them.
Also, it is said that whenever a new user gets on your website, you got 4 seconds to convince that person to go forward and visit other pages of your site as well. So, my opinion is to stop thinking about that old flash, unless you only target desktop users, but I pretty much doubt you do.
Besides this, if flash is still used, it does not mean you have to use it. Mamy they are still using it because they do not know how to develop a site using HTML5, or they simply do not care because they are big enough, or whatever.
So my opinion is that you firstly define your target audience, take in consideration mobile devices usage, and instead of that flash, just use HTML5 or Gif if it is something small. Note that HTML5 is a resource consumer (for users), I only recommend it on small sites, like presentation sites, not on webshops.
I hope this helped.