I've had this happen on a few occasions but it didn't seem to affect anything. It tends to happen because the pea straw has been stored wrongly, probably rained on while in the bale and since you've had rain, it's triggered the spores. Be sure to buy the freshest baled straw you can find, preferably stored under cover. It might pay to fluff up your mulch a bit and get the air in on a dry day. Putting it down in 'biscuits' isn't the best way, it does better if there's a bit of air circulation.Dry heat should reduce the mould. I've just done my vegetable beds with pea straw too and we bought it in the paddock minutes after it was baled, nice and loose and airy.

I deliberately wet my bales of pea straw and leave them to sit for a few weeks to go mouldy before i open them and use them for mulch. I find the mulch breaks down into rich compost more quickly if i do this.
I have never heard of a mould that has tendrils - perhaps you just have a few weeds growing?
Always best to use a face mask if you follow my example. You can get a lung full of spores from the mould that grows.
Cheers, Jan