Currently the best access is the Bellefields trailhead on 118th Ave SE from the gravel parking lot next to it.
In my opinion the easiest place to find & park of you've never been there is the Winters House parking lot, which is just north of the Park & Ride on Bellevue Way [address is 2012 Bellevue Way SE]. From there you can easily find the boardwalk part of the trails. There's a good map available at parks.bellevuewa.gov (search Enatai Beach Park and the Mercer Slough Water Trail) that shows the trails, Winters House, and the other possible parking areas others mentioned, like the Environmental Education Center, or along the street on 118th.
I go in and park off of coal creek
Search for park and ride (P&R). It's attached to the bus stop.
I always go in from Mercer Slough Blueberry Farm but I think that's closed due to the construction. Looks like there is an entrance at Mercer Slough Environmental Education Center, but there is only handicap parking, although in Google Street View there looks like plenty of close street parking.
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