не мога да видя къде ми е грешката

Няма изискване думите в двойката да са с еднаква дължина.
Проиграйте си кода с "@fff@@fffn@". Резултата не е ок.

Имате поне една грешка при "firstPart" и "secondPart"; още: тяхното присъствие в кода е грешка само по себе си - то не е необходимо; пробвайте (класа пада, "тонове" код изчезват): 

foreach (Match match in wordPairs)
{
    var ms = match.ToString ();
    if (ms.IsPalindrome ())
        mirrorWords.Add (ms.Trim (new char[] { '#', '@' }).Replace ("##", " <=> ").Replace ("@@", " <=> "));
}

Регулярен израз който намира двойките (можете да го ползвате за еталон при тестване):

grep -o '\([@#]\)[A-z]\{3,\}\1\1[A-z]\{3,\}\1'

Втората част от условието касае палиндром. Пробвайте:

internal static bool IsPalindrome(this string value)
{
    if (string.IsNullOrEmpty (value)) return false; // Empty string policy
    for (int i = 0, j = value.Length - 1; i < value.Length / 2; i++, j--)
        if (value[i] != value[j]) return false;
    return true;
}

Когато wordPairs.Count <= 0, какъв точно е смисъла да проверявате за палиндроми?
В момента за вход "fsdfsdlk" програмата Ви ще отпечата:
"No word pairs found!"
"No mirror words!"
А дали трябва?

Вашия код с малко мои кръпки + мое решение + автоматизиран тест от 100000 различни входни стринга:

internal static class MyStringExtensions
{
    /// <summary>When the string is null or empty returns false; otherwise returns true when the string is a palindrome</summary>
    internal static bool IsPalindrome(this string value)
    {
        if (string.IsNullOrEmpty (value)) return false; // Empty string policy
        for (int i = 0, j = value.Length - 1; i < value.Length / 2; i++, j--)
            if (value[i] != value[j]) return false;
        return true;
    }
}

class Program
{
    // Generates random input; use "expected" to compare the output of your Analyze()
    static string Gen(ref List<string> expected)
    {
        // generate random string of random length filled with the allowed symbols
        const string SYM = "#@abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ";
        Random rnd = new Random ();
        StringBuilder s = new StringBuilder (rnd.Next (104));
        for (int i = 0; i < s.Capacity; i++) s.Append (SYM[rnd.Next (SYM.Length)]);

        // put random number of pairs and palindromes in it
        StringBuilder palidromes = new StringBuilder ();
        int pairs = rnd.Next (5), actual_pairs = 0, pos = 0;
        for (int i = 0; i < pairs; i++)
        {
            int rnd_ofs = rnd.Next (5);
            // gen the pair
            int rnd_w1_len = 3 + rnd.Next (5);
            int rnd_w2_len = 3 + rnd.Next (5);
            StringBuilder words_builder = new StringBuilder (rnd_w1_len + rnd_w2_len);
            for (int j = 0; j < words_builder.Capacity; j++) words_builder.Append (SYM[2 + rnd.Next (SYM.Length - 2)]);
            var words = words_builder.ToString ();
            var word1 = words.Substring (0, rnd_w1_len);
            var word2 = words.Substring (rnd_w1_len, rnd_w2_len);
            bool palindrome = 0 == rnd.Next (5) % 2;
            char rnd_sep = SYM[rnd.Next (2)];
            string pair = string.Format ("{0}{1}{0}{0}{2}{0}", rnd_sep, word1,
                (palindrome ? new string (word1.Reverse ().ToArray ()) : word2));
            // is there space left
            pos += rnd_ofs;
            if (pos + pair.Length > s.Length) break;
            //
            if (palindrome) palidromes.Append (FormatPalindrome (pair) + ", ");
            actual_pairs++; s.Remove (pos, pair.Length); s.Insert (pos, pair); pos += pair.Length;
        }
        if (0 == actual_pairs) expected.Add ("No word pairs found!");
        else
        {
            expected.Add (string.Format ("{0} word pairs found!", actual_pairs));
            if (palidromes.Length <= 0) expected.Add ("No mirror words!");
            else
            {
                expected.Add ("The mirror words are:");
                Debug.Assert (palidromes.Length > 3 + 3 + 4, "Bug: min pair length is 10");
                palidromes.Remove (palidromes.Length - 2, 2);
                expected.Add (palidromes.ToString ());
            }
        }
        return s.ToString ();
    }

    static string FormatPalindrome(string value)
    {
        return value.Trim (new char[] { '#', '@' }).Replace ("##", " <=> ").Replace ("@@", " <=> ");
    }

    // Your solution patched as described.
    static List<string> Analyze_YourCodeWithMyPatchesEdition(string input)
    {
        List<string> result = new List<string> ();

        const string pattern = @"(?<separate>[@#]{1})[A-Za-z]{3,}\k<separate>{2}[A-Za-z]{3,}\k<separate>";
        List<string> mirrorWords = new List<string> ();
        MatchCollection wordPairs = Regex.Matches (input, pattern);
        foreach (Match match in wordPairs)
        {
            var ms = match.ToString ();
            if (ms.IsPalindrome ())
                mirrorWords.Add (ms.Trim (new char[] { '#', '@' }).Replace ("##", " <=> ").Replace ("@@", " <=> "));
        }
        if (wordPairs.Count > 0)
        {
            result.Add (string.Format ("{0} word pairs found!", wordPairs.Count));
            if (mirrorWords.Count > 0)
            {
                result.Add ("The mirror words are:");
                result.Add (string.Join (", ", mirrorWords));
            }
            else
                result.Add ("No mirror words!");
        }
        else
            result.Add ("No word pairs found!");
        return result;
    }

    // My solution
    static List<string> Analyze(string input)
    {
        List<string> result = new List<string> ();

        const string pattern = @"(?<separate>[@#])[A-z]{3,}\k<separate>{2}[A-z]{3,}\k<separate>";
        if (string.IsNullOrEmpty (input)) throw new Exception ("Wrong input");
        var wordPairs = Regex.Matches (input, pattern);
        if (wordPairs.Count <= 0)
            result.Add ("No word pairs found!");
        else
        {
            result.Add (string.Format ("{0} word pairs found!", wordPairs.Count));
            var palindromes = wordPairs.Cast<Match> ().Select (m => m.ToString ()).Where (w => w.IsPalindrome ()).ToArray ();
            if (palindromes.Length <= 0) result.Add ("No mirror words!");
            else
            {
                result.Add ("The mirror words are:");
                StringBuilder p = new StringBuilder ();
                p.Append (FormatPalindrome (palindromes[0]));
                for (int i = 1; i < palindromes.Length; i++)
                    p.Append (", " + FormatPalindrome (palindromes[i]));
                result.Add (p.ToString ());
            }
        }

        return result;
    }

    static void Main()
    {
#if TEST
        // Automated test
        Stopwatch sw = new Stopwatch ();
        sw.Start ();
        List<string> expected = new List<string> ();
        const int TEST_ITERATIONS = 100000;
        for (int t = 0; t < TEST_ITERATIONS; t++)
        {
            expected.Clear ();
            Console.Write (string.Format ("Test {0}/{1}\r", t + 1, TEST_ITERATIONS));
            var input = Gen (ref expected);
            var actual = Analyze_YourCodeWithMyPatchesEdition (input);
            //var actual = Analyze (input);
            for (int i = 0; i < expected.Count; i++)
                if (i >= actual.Count || actual[i] != expected[i])
                {
                    Console.WriteLine ("Bug:");
                    Console.WriteLine ("Input:\"" + input + "\"");
                    Console.WriteLine ("Expected:"); foreach (var l in expected) Console.WriteLine (l);
                    Console.WriteLine ("Actual:"); foreach (var l in actual) Console.WriteLine (l);
                    break;
                }
        }
        sw.Stop ();
        Console.WriteLine ();
        Console.WriteLine ("Time: " + sw.ElapsedMilliseconds + " ms");
#endif
        foreach (var l in Analyze_YourCodeWithMyPatchesEdition (Console.ReadLine ())) Console.WriteLine (l);
        Console.ReadLine ();
    }
}// class Program