We have a great facebook group for home educators in our county. It was started by just one person (who lives quite a long way from me), and it was set up so that any member could add any other home educating member. Within months it spiralled and there are now nearly 150 members on there. Anyone can post an event.
If you know even one other home educating family, you could start such a thing, invite them to join, and get them to invite others. The key is to make it a closed group but make everyone an admin member. Then anyone can add anyone else.
Once you have even one or two people, start posting events. This can be "picnic at brokemum's house" or if your son is interested in something like art, "painting day at brokemums house". Send out an event invite to all the members of the group. Most won't come, sometimes maybe no one will come, but keep going. If the group grows you only need one or two people to turn up and you've done what you wanted. if you have anything you're good at you can offer - book club, art club, music hour, drama session, you'll tend to pull in lots of people who don't feel confident with those things themselves.
In our group it's acceptable to ask a small amount of money (50p - £1, maybe) to cover your costs, particularly if you are supplying art materials etc. Get people to bring their own food - one thing I like about home ed is that you can invite masses of people over for the day and no one expects you to feed them! Make sure you keep any costs to visitors as low as possible, because you'll find lots of home edders are like you and will be put off if it is too much.
It's a bit of effort but well worth it. The lady who started our group did so for very similar reasons to you - she was fed up with taking her children to the park alone. Now she posts park meetups about 3 times a week and always seems to have a crowd going, and there ar about 5 events a week on there, posted by lots of different people.