This answer to another question makes me wonder: Did anointed kings of Israel have any contact with their parents? David was dead when Shlomo succeeded him, and Shlomo ruled until his own death. But were Shaul and David's fathers alive during their respective reigns? What about the mothers of all of the above? What about later kings (assuming they had legitimacy)?
After perusing the Tanach to confirm, I have some found some answers to this question:
First, Shlomo definitely had contact with both his parents. He was anointed by Nathan and Zadok, at David's command, in I Melachim 1:32-34. There must have been some contact time between then and when David charged him with farewell instructions in Chapter 2. As for Bathsheba, after David's death, she speaks to him for Adonyahu, and Shlomo gives her a throne next to his, in 2:19.
As for later kings, first of all, anointed and reigning King Asa must have had contact with his mother Queen Ma'acah--he deposes her for idolatry in I Melachim 15:13. Later on, the story of soon-to-be King Yehu in II Melachim 9 shows that King Yoram's mother Yezba'al was alive and active while he was king. The same goes for the other slaughtered king--Judean King Ahazyahu's mother Atalyahu was very obviously alive and active during his short reign-- II D. Hayamim 22:3 even says she was his counsellor in wickedness. Some years later, II Melachim 15:5 seems to indicate a co-regency of Azaryahu/Uziyahu(after he was smitten with leprosy) and his son Yotham. The last instance of contact between a ruling King and a parent appears to be at II Melachim 24:12, where King Yehoyachin/Conyahu and his mother Nehushta get taken to Babylon.
While I don't know how it reconciles with the issue you referred to, we see that although Shlomo HaMelech is appointed king (at Dovid HaMelech's command) in 1 Kings 1:32-40 prior to the passing of Dovid HaMelech. In verse 43-4 we read "Jonathan answered and said to Adonijah, 'To the contrary, our lord King David has made Solomon King!...Furthermore, Solomon has sat upon the royal throne." (Artscroll translation). Afterwards in 1 Kings 2 we see Dovid haMelech giving his son the King his final advise and instruction. As Jake mentioned 1 Kings 2:19 Bathsheba had a throne next to her son Shlomo HaMelech.
The Medrashim say Shlomo's Mother beat him for waking up late the day after his wedding to queen Sheba as she tricked him by placing a Sheet over their bed to make him believe it was still night and this was the first day the Avodah in the Beis Hamikdash was supposed to start. Mentioned here in the Jewish Encyclopedia-under Solomon's Marriage