Hi and welcome to SilverStripe.
Embedded content like SWFs (Flash) or Movies always start anew with every page-refresh.
There are several workarounds for this, but they are either complicated or a SEO nightmare.
Here the possibilities I can think of:
1) Create a frameset. Run the flash header in the top frame and the rest of your site in the frame below.
Pros: Easy to do
Cons: Really bad in terms of search engine friendliness, bookmarking etc.
2) Load your site-contents via AJAX instead of reloading the whole site
Pros: No frameset, other elements (like the navigation) could stay "static" as well
Cons: Almost the same problems as with frames. Problem can be solved by using a deep-linking script like SWFAddress
3) Keep track of the flash-position inside flash (by storing the "playhead-position" in a shared object or similar and restart the movie from there at every page-refresh)
Pros: Website can be built as usual
Cons: Requires some advanced flash skills and most importantly: Access to the flash source.