I use this code to login:
CookieCollection cookies = new CookieCollection();
HttpWebRequest request = (HttpWebRequest)WebRequest.Create(\"example.com\");
req         
        There are some recommendations here: Using CookieContainer with WebClient class
However, it's probably just easier to keep using the HttpWebRequest and set the cookie in the CookieContainer:
The code looks something like this:
 // Create a HttpWebRequest
HttpWebRequest request = (HttpWebRequest)WebRequest.Create(getUrl);
// Create the cookie container and add a cookie
request.CookieContainer = new CookieContainer();
// Add all the cookies
foreach (Cookie cookie in response.Cookies)
{
    request.CookieContainer.Add(cookie);
}
The second thing is that you don't need to download the site again, since you already have it from your web response and you're saving it here:
HttpWebResponse getResponse = (HttpWebResponse)getRequest.GetResponse();
using (StreamReader sr = new StreamReader(getResponse.GetResponseStream(), Encoding.GetEncoding("windows-1251")))
{
        webBrowser1.DocumentText = doc.DocumentNode.OuterHtml;
}
You should be able to just take the HTML and parse it with the HTML Agility Pack:
HtmlDocument doc = new HtmlDocument();
doc.LoadHtml(webBrowser1.DocumentText);
And that should do it... :)
Try caching cookies from previous response locally and resend them each web request as follows:
private CookieCollection cookieCollection;
...
    parserObject = new HtmlWeb
                {
                    AutoDetectEncoding = true,
                    PreRequest = request =>
                    {
                        if (cookieCollection != null)
                            cookieCollection.Cast<Cookie>()
                                .ForEach(cookie => request.CookieContainer.Add(cookie));
                        return true;
                    },
                    PostResponse = (request, response) => { cookieCollection = response.Cookies; }
                };
Check HtmlAgilityPack.HtmlDocument Cookies
Here is an example of what you're looking for (syntax not 100% tested, I just modified some class I usually use):
public class MyWebClient
{
    //The cookies will be here.
    private CookieContainer _cookies = new CookieContainer();
    //In case you need to clear the cookies
    public void ClearCookies() {
        _cookies = new CookieContainer();
    }
    public HtmlDocument GetPage(string url) {
        HttpWebRequest request = (HttpWebRequest)WebRequest.Create(url);
        request.Method = "GET";
        //Set more parameters here...
        //...
        //This is the important part.
        request.CookieContainer = _cookies;
        HttpWebResponse response = (HttpWebResponse)request.GetResponse();
        var stream = response.GetResponseStream();
        //When you get the response from the website, the cookies will be stored
        //automatically in "_cookies".
        using (var reader = new StreamReader(stream)) {
            string html = reader.ReadToEnd();
            var doc = new HtmlDocument();
            doc.LoadHtml(html);
            return doc;
        }
    }
}
Here is how you use it:
var client = new MyWebClient();
HtmlDocument doc = client.GetPage("http://somepage.com");
//This request will be sent with the cookies obtained from the page
doc = client.GetPage("http://somepage.com/another-page");
Note: If you also want to use POST method, just create a method similar to GetPage with the POST logic, refactor the class, etc.