Not sure why your login doesn't work when you try to login the second time. As for being "logged out" you're probably not really being logged out, you're probably viewing the site on a different URL than the one you logged in with originally. I noticed your site answers to both www.parentjobnet.org and parentjobnet.org but neither redirects to the other (i.e., parentjobnet.org redirects to www.parentjobnet.org). This means that your site is seen as two different sites by browsers (and search engines for that matter) so when you login at parentjobnet.org, a cookie is set by your browser for that domain. When you go to a page (either by direct entry or a link) at www.parentjobnet.org, the cookie set by browser for parentjobnet.org isn't valid and therefore you are not logged in. If you have your browser set to remember passwords, it could have different passwords saved for www.parentjobnet.org and parentjobnet.org. This would explain why you are able to login on one and not the other.
To get past this dueling url issue, see the "Forcing a Preferred Host Name" article.
Hope this helps,