What’s more important, keeping Shabbat or Kibbud Av V’em?

Question

In a situation where these two conflict, which trumps the other?

Answer 1

Shabbat.

Leviticus 19:3:

Every man shall fear his mother and his father, and you shall observe My Sabbaths. I am the Lord, your God. גאִ֣ישׁ אִמּ֤וֹ וְאָבִיו֙ תִּירָ֔אוּ וְאֶת־שַׁבְּתֹתַ֖י תִּשְׁמֹ֑רוּ אֲנִ֖י יְהוָֹ֥ה אֱלֹֽהֵיכֶֽם:

Rashi cites the Talmud, Bava Metzia 32b:

“Although I have admonished you regarding the fear of your father, nevertheless, if he tells you to desecrate the Sabbath, do not listen to him.” וְאֶת־שַׁבְּתֹתַי תִּשְׁמֹרוּ: סָמַךְ שַׁבָּת לְמוֹרָא אָב, לוֹמַר אַף עַל פִּי שֶׁהִזְהַרְתִּיךָ עַל מוֹרָא אָב, אִם יֹאמַר לְךָ חַלֵּל אֶת הַשַּׁבָּת, אַל תִּשְׁמַע לוֹ, וְכֵן בִּשְׁאָר כָּל הַמִּצְווֹת (בבא מציעא ל"ב):

Thus the verse means "honor your parents, but what's even more important, keep Shabbat."

There can be shades of grey if it's a conflict between, say, the parents' needs and something preferred by halacha about keeping Shabbat; if someone needs to take care of a parent they might have to pray at home and miss synagogue, for example; often something can be worked out to accommodate both. (See more here.) But when push comes to shove, if the parent demands a black-on-white violation of Shabbat, the answer is "no."

Even then, it's important to try and maintain a respectful relationship; "I'm sorry but I can't do that", not "I'm so much smarter than you and you're a terrible sinner."

Answer 2

The Talmud says that if you must choose between doing two mitzvot, do first the one that recurs more frequently:

תָּדִיר וְשֶׁאֵינוֹ תָּדִיר — תָּדִיר קוֹדֵם [Berakhot 27a and 51b, Pesachim 114a, Megillah 29b, Sukkah 54b, Horayot 13, and Zevachim Mishna 10:1]

For example, we put on tzitzit (that is, the tallit) before we put on tefillin, because tzitzit recurs every day, but tefillin recurs every day except Shabbat. [Sefer Ta'amei Haminhagim Umkorei Hadinim]

As @Shalom said, you don't violate Shabbat because a parent asks you to. However, if you have to decide which commandment to follow first, I would conclude that you must first tend to the needs of your parents, because that commandment is always present, but Shabbat is only one day out of seven.