Find pairs in binary search tree in which sum of nodes keys is equal key
I just have minor nitpicks.

1

You can simplify the methods for computing a successor/predecessor:

public static Node successor(Node node) {
    if (node.right != null) {
        return min(node.right);
    }

    while (node.parent != null &amp;&amp; node.parent.right == node) {
        node = node.parent;
    }

    return node.parent;
}


Also, you compute the min(node.right) above in code; hence, you repeated yourself.

2

I would rip off the implementation of your sum pair algorithm out of main().

3

I would have a class, say, NodePair, which simply stores a pair of nodes.

4

The actual algorithm could just return the list of those NodePair objects.

5

Since you are printing the results, you could override toString() in NodePair and make it return a string in the format that meets your needs.

6 

inOrder is not used. In case you don't need it either, consider removing it.

7

You import java.lang, yet you don't need to: it is imported by default. Also, you left hanging around two packages (java.util and java.io), remove them. Finally, I would encourage avoiding wildcard imports: import on individual basis, let people see what classes you use in your source.

Summa summarum

All in all, I had this in mind:

import java.util.ArrayList;
import java.util.List;

class Main {

    static class Node {

        Node(Node left, Node right, Node parent, int key) {
            this.left = left;
            this.right = right;
            this.parent = parent;
            this.key = key;
        }

        public Node left;
        public Node right;
        public Node parent;

        public int key;

        @Override
        public String toString() {
            return new StringBuilder("Node ").append(key).toString();
        }
    }

    public static Node successor(Node node) {
        if (node.right != null) {
            return min(node.right);
        }

        while (node.parent != null &amp;&amp; node.parent.right == node) {
            node = node.parent;
        }

        return node.parent;
    }

    public static Node predeccessor(Node node) {
        if (node.left != null) {
            return max(node.left);
        }

        while (node.parent != null &amp;&amp; node.parent.left == node) {
            node = node.parent;
        }

        return node.parent;
    }

    public static Node min(Node node) {
        while (node.left != null) {
            node = node.left;
        }

        return node;
    }

    public static Node max(Node node) {
        while (node.right != null) {
            node = node.right;
        }

        return node;
    }

    public static final class NodePair {
        public Node node1;
        public Node node2;

        public NodePair(Node node1, Node node2) {
            this.node1 = node1;
            this.node2 = node2;
        }

        @Override
        public String toString() {
            return new StringBuilder("[").append(node1)
                                         .append(", ")
                                         .append(node2)
                                         .append("]")
                                         .toString();
        }
    }

    public static List&lt;NodePair&gt; findPairsWithSum(Node root, int sum) {
        List&lt;NodePair&gt; nodePairList = new ArrayList&lt;&gt;();
        Node minnode = min(root);
        Node maxnode = max(root);

        while (minnode.key &lt;= maxnode.key &amp;&amp; minnode != maxnode) {
            if (minnode.key + maxnode.key == sum) {
                nodePairList.add(new NodePair(minnode, maxnode));
                minnode = successor(minnode);
                maxnode = predeccessor(maxnode);
            } else if (minnode.key + maxnode.key &lt; sum) {
                minnode = successor(minnode);
            } else {
                maxnode = predeccessor(maxnode);
            }
        }

        return nodePairList;
    }

    public static void main(String[] args) throws java.lang.Exception {
        Node root = new Node(null, null, null, 5);//      5
        Node a = new Node(null, null, null, 2);   //     / \
        Node b = new Node(null, null, null, 1);   //    2   8
        Node c = new Node(null, null, null, 3);   //   /\    \
        Node d = new Node(null, null, null, 8);   //  1  3    9
        Node e = new Node(null, null, null, 9);   //

        root.left = a;
        root.right = d;
        a.parent = root;
        d.parent = root;

        a.left = b;
        a.right = c;
        b.parent = a;
        c.parent = a;

        d.right = e;
        e.parent = d;

        // Print all the node pairs summing to 10.
        for (NodePair pair : findPairsWithSum(root, 10)) {
            System.out.println(pair);
        }
    }
}


Hope that helps.

Edit

Actually, what comes to printing the node pairs, I realized that it makes sense to override Node.toString() as well. After that, building a string representation of a NodePair simplifies a bit. Please, for details refer to Node.toString and NodePair.toString.