Answer. This sounds to me a problem that transcends the park rules. Based upon your description of these circumstances, you are certainly within your rights to issue a 30-day curable notice under ORS 90.630(1)(a) and (b) for violation of one or more of your park rules (I'd have to read them to say for certain), the rental agreement (same caveat), or ORS 90.740(4)(j) (Tenant Obligations).
You have not mentioned whether the Temporary Occupants are staying there under a written Temporary Occupant Agreement. If so, it is far easier to terminate them, than terminating the tenancy of a tenant who is occupying the space under a signed Rental Agreement.
But the larger problem, from a humanitarian point of view, is what happens after you have all of the offenders removed? The cycle will likely repeat itself somewhere else, and the situation could become even worse for the children. I suggest you have a discussion with the tenants and make it clear that (a) the extra occupants will have to leave, and that (b) some effort should be made to assist the mother and children transition into a stable living environment. Perhaps a little advance research is in order for you, since it sounds as if the children's best interests are not being properly addressed by anyone in the home. See, https://www.oregon.gov/DHS/Offices/Pages/Child-Welfare.aspx
 Behave, and require persons on the premises with the consent of the tenant to behave, in a manner that does not disturb the peaceful enjoyment of the premises by neighbors.