Dragging/harrowing to spread manure will only help kill strongyle eggs if it’s hot and dry enough. Getting them out of the manure does remove their nice wet setting, but then can survive quite a while if it’s cool enough, even if it’s dry. For that reasons you generally don’t want to drag when it’s going to be less than 85* for a while
Unfortunately, no amount of cold (here in the continental US anyway) kills strongyle eggs.
the “don’t let them back on it for a month” is coming from the idea that if you drag a pasture as horses are coming off, that month gives time for parasite eggs to be killed (see above), and grass to regrow to a healthier height before being allowed to be grazed again.
that said, I don’t have the real option to remove horses from pasture for that long, and it’s Winter when I need to drag most often. Dung beetles do a really good job keeping manure in the pasture to a minimum during the late Spring to early Fall months. It’s a risk I take, spreading strongyle eggs, but they’re also eating hay and not doing a lot of grazing. Regular clean/low FECs tell me that my horses have healthy enough immune systems, and I deworm appropriately in Spring and Fall, that the risk isn’t outweighing the benefits of not having a field full of manure piles.
Mowing for healthy grass height will ignore most manure piles, maybe at best knock the top off them.
If you’re “harrowing to prep for broadcast seeding”, and by “harrow” you mean drag, then I would do it in reverse. Seed, mow, drag, to get the best seed to ground contact, and provide some light protection against birds. You can also mow, seed, drag if the grass is too tall to get seeds close enough to the ground to start.