Men get called dickhead, bellend, nob, prick, manbaby, manchild, etc all the time on Mumsnet. They also get called cunts and twats quite a lot.
I personally don't see calling someone a cunt as being any different from calling someone a dick. 'Private' body parts are routine insults for both sexes - cunt, twat, arsehole, tit, dick, prick, cock etc - and I don't find the use of cunt or twat as insults to be misogynistic. Context is always key, of course. For example, some of my favourite body part related insults are also the words I would use appreciatively for those body parts in the bedroom.
'Bitch' and 'cow' are obviously gendered insults because they refer to female animals. I'm not sure I necessarily find it more offensive to insult someone by comparing them to animal of their own sex than it is to use a generic insult, but I can definitely see why people might feel that way and I'm certainly open to being more mindful of that.
I do think it's extremely misogynistic to call a woman a slag, slapper, tart, skank etc, though. It implies that women should be judged for enjoying as much consensual sex as they want, and for not living up to some patriarchal ideal of purity.
I think the most misogynistic insults on Mumsnet are - ironically - the ones that imply anyone who doesn't share a very specific set of views on topics of sexual behaviour, gender identity or various other things must somehow only hold those views because she's desperate to be accepted by men. 'Pick-me', 'cool wife', 'handmaiden' etc. All massively misogynistic, despite invariably being used by women who call themselves feminists.
Also misogynistic: the posters who immediately say 'you must be a man' to anyone who expresses a different view on sex and relationships to theirs.