Can only answer from a UK perspective, sorry. I think it is harder for dads. There's quite a few on the parent circuit round our way, but i must say i have only perhaps twice invited a dad and his kid(s) back to mine for coffee as I did for lots and lots of mums. It's getting a bit easier as kids get older though.
Unpicking that, there were several reasons - I always felt/feel embarrassed at the 'exposure' of the kind of chat I go in for with other mums to the male gaze - i just assume the guy thinks it is pretty vapid - a lot of other mums would too, but when you and the other mums perhaps don't know each other that well, you tend to do a lot of lowest-common denominator stuff - yucky post-childbirth stuff initially, then how the kids are doing (not competitive I hope), current favourite children's books, sometimes 'what a lovely top, really suits your dd', all the stuff that other women as well as men complain about on here which means they hate parent/toddler groups. I don't mind it, in fact I find it comforting and easy, and many of these relationships have developed so that we can talk about much more of a range of stuff. However, whenever a man is there too I clam up and feel they must think I/we are brainless. I therefore would not TBH automatically invite a dad to a group thing, and I do feel a little bit cautious about inviting a dad to my house on his own with kids in case he thinks I am making a pass and recoils in horror!
So having said all that, the best possible answer IMO is to invite people to stuff you want to do, and keep chipping away at it. I would say it took at least two years in the UK (in an area I moved to when pregnant) to have a social circle, for people to actually ring me and invite me to anything at all, rather than me making 90% of the running. Don't assume it has to be other dads, although you may get a better response from women if you invite them to take the kids into town/for a walk/to a gallery/for ice cream/soft play etc etc. Ask for numbers and write down names, names of children, things about them that will jog your memory ('Jane, translator, 2 kids, partner SAHD wd like to meet up, tues best day, phone 234398' sort of thing).
The most important thing is not to expect more than one aspect of the person to click with you - if they are kind, and happy to have you over, and you can find SOMETHING to talk about, that's a good start; if they are funny, and love to meet for a great boozy night, well great, even if they are not interested in meeting with children. Persevere, don't expect bezzy mates on every corner.
Also I am still after all this time scared of intruding into cliques that don't want me, so someone I have a big chat with one week I may only smile at next week - prob v frustrating for people but I am petrified of overstepping the mark. Snot easy but it's better than the alternative.