I Samuel 16:11 and Psalms 118:22 imply that David was a rejected child in his family.
וַיֹּ֨אמֶר שְׁמוּאֵ֣ל אֶל־יִשַׁי֮ הֲתַ֣מּוּ הַנְּעָרִים֒ וַיֹּ֗אמֶר ע֚וֹד שָׁאַ֣ר הַקָּטָ֔ן וְהִנֵּ֥ה רֹעֶ֖ה בַּצֹּ֑אן וַיֹּ֨אמֶר שְׁמוּאֵ֤ל אֶל־יִשַׁי֙ שִׁלְחָ֣ה וְקָחֶ֔נּוּ כִּ֥י לֹא־נָסֹ֖ב עַד־בֹּא֥וֹ פֹֽה׃
Then Samuel asked Jesse, “Are these all the boys you have?” He replied, “There is still the youngest; he is tending the flock.” And Samuel said to Jesse, “Send someone to bring him, for we will not sit down to eat until he gets here.”
אֶ֭בֶן מָאֲס֣וּ הַבּוֹנִ֑ים הָ֝יְתָ֗ה לְרֹ֣אשׁ פִּנָּֽה׃
The stone that the builders rejected has become the chief cornerstone.
David was an afterthought amongst Yishai's sons, and is called the stone that the builders rejected (Targum and Metzudas Dovid ad. loc. Cf. Rashi). The builders seem to be referring to his family (see Pesachim 119a and Rashi ad. loc. (ד"ה אבן מאסו)). Why would this be?
There’s a Midrash which explains1 that David had a very sketchy conception. Yishai, David’s father, had been separated from his wife for three years2. His wife had a maidservant, and Yishai wanted to have more children3. He therefore petitioned her for her cooperation4. She wasn’t interested, and asked her mistress, Yishai's wife, for advice. Yishai’s wife came up with a plan: the maidservant will pretend to be interested, but in the dark she’ll switch with her mistress. This is what happened, and Yishai’s wife became pregnant with David.
Since Yishai didn’t know he had been intimate with his own wife, he and his other sons thought she had committed adultery. David would therefore have the status of a mamzer5. This could explain why his father "abandoned" him. Not sure about his mother, who knew the truth. Maybe she played along with her husband.
PS - David’s status was finally cleared among his family when he was chosen by the prophet Shmuel to become the next King of the Jews6.
Note: I'm not proving that Psalms Chapter 27 specifically is referring to King David's parents having abandoned him. I'm merely showing the understanding of the OP to be a discussion which does exist.
1 Yalkut Me’am Loez to I Samuel 16:11. This story is mentioned in many sources, one of which is Kli Yakar ad. loc. who says he heard it from Rav Shlomo Alkabetz. Rav Alkabetz himself says it’s from a Midrash, the earliest I found being in Yalkut Machiri ad. loc. (c. 14th century), quoting an unnamed Midrash to Psalms 118:19
2 The Rama MiPano in his Asara Ma’amaros Ma’amar Chikur HaDin 3:10 explains that Yishai was concerned that since the Torah forbids marrying someone from Moav (Deuteronomy 23:4), his ancestor Boaz should have been forbidden from marrying Ruth (see Ruth 4:13). There had been a debate raging ever since then if the Torah only forbade male members from Moav, or even female. Boaz had ruled the former, and married Ruth. Even though that was the decided ruling (see Yevamos 77a), Yishai, due to his great righteousness, was concerned that perhaps the halacha was not so. He would consequently be forbidden from marrying into the Jewish people. Therefore, he separated physically from his wife.
3 Asara Ma’amaros loc. cit.; this was to fulfill ולערב אל תנח ידך (see Yevamos 62b). Yalkut Machiri only says that Yishai desired his wife’s maidservant
4 Asara Ma’amaros loc. cit. explains that Yishai planned to make a stipulation: If the halacha is that only male members of Moav are forbidden to marry into the Jewish people, then the maidservant should become free and a full member of the Jewish people. He would thus marry her. If, however the halacha is that even female members of Moav are forbidden, then she should remain a maidservant, as he would be allowed to be with her in her present state
5 This is the explanation of Yalkut Me’am Loez loc. cit. However, both Asara Ma’amaros and Kli Yakar loc. cit. sound like Yishai eventually knew he had impregnated his wife. They explain that the reason why David was the rejected child was because he had the status of a ben temurah (see Nedarim 20b), as Yishai was thinking of another woman while he was intimate with his wife. Although it would be tempting to say that the Yalkut Me’am Loez means like "Rashi" to Nedarim loc. cit. (ד"ה בני תמורה), who explains that benei temurah are almost considered mamzerim, this would be incorrect. He writes explicitly that they thought he was a mamzer since Yishai had been separated from his wife. Cf. Yalkut Machiri, who only mentions that they rejected David because he was completely red.
6 Yalkut Me’am Loez loc. cit.