Head up the small road that crosses behind the Cotswold outlet shop at the South end of B-Y-C (past the Vagabond Bunkhouse). This is a pleasant ride that follows the river on very quiet roads (also a cycle route of some sort). It brings you out onto the A470 after it makes a dog-leg under the railway. There’s a cycle path of sorts along the A470. This will take you to the bridge you’ve seen after about 500m.
I haven’t ridden up to Penmachno from there myself, but know people who have – They report that it becomes very disorientating in the forest – there is a maze of fire roads, some of which appear to be going in the correct direction, but end up in dead ends. Their advice was to plan a route using google earth / Strava, etc. It’s pretty steep from what I understand, and retracing your path gets tiresome quite quickly.