Heb. 1:8 is a quotation of Psa. 45:7 (v. 8 according to Masoretic verse numbering).
In Psa. 45:7, it is written,
אָהַבְתָּ צֶּדֶק וַתִּשְׂנָא רֶשַׁע עַל כֵּן מְשָׁחֲךָ אֱלֹהִים אֱלֹהֶיךָ שֶׁמֶן שָׂשׂוֹן מֵחֲבֵרֶךָ
The word in question is מֵחֲבֵרֶךָ, which consists of the prepositional -מ prefixed to the word חֲבֵרֶךָ, meaning "your (sg.) companions/fellows."
The prepositional -מ can be used in Hebrew to indicate comparison, i.e., "...more than..."
For example, in Hebrew Grammar, 2nd ed., p. 429, §133a (1), Heinrich Friedrich Wilhelm Gesenius wrote,
Hebrew possesses no special forms either for the comparative or superlative of the adjective. In order to express a comparative, the person or thing which is to be represented as excelled in some particular quality is attached to the attributive word by the preposition מִן־ (מִ·), e.g. 1 S 92 גָּבֹהַּ מִכָּל־הָעָם higher than any of the people. The fundamental idea evidently is, tall away from all the people (beyond all the people); cf. Ju 1418 מַה־מָּתוֹק טִדְּבַשׁ וּמֶה עַז מֵֽאֲרִי what is sweeter than honey? and what is stronger than a lion? Ez 283, Am 62 Frequently an infinitive appears as the object of the comparison, e.g. Gn 2919 it is better that I give her to thee, than that I should, give her, &c.; Ex 1412, ψ 1188f.
Therefore, the word מֵחֲבֵרֶךָ simply means "more than your companions/fellows." God anointed God with the oil of gladness more than his companions/fellows.
Another example can be found in Deu. 30:5 where the word מֵאֲבֹתֶיךָ means "more than your fathers."
As far as παρά being used in a comparative clause, Thayer (p. 478) notes,
b. above, beyond: παρὰ καιρὸν ἡλικίας, Heb. 11:11; παρʼ ὃ δεῖ (Plut. mor. p. 83 f. [de profect. in virt. § 13]), Ro. 12:3; i. q. more than: ἁμαρτωλοὶ παρὰ πάντας, Lk. 13:2; ἔχρισέ σε ἔλαιον παρὰ τοὺς μετ. more copiously than [A. V. above] thy fellows, Heb. 1:9 (fr. Ps. 44 (45):8; ὑψοῦν τινα παρά τινα, Sir. 15:5); κρίνειν ἡμέραν παρʼ ἡμέραν, to prefer one day to another (see κρίνω, 2), Ro. 14:5. Hence it is joined to comparatives: πλέον παρά τ. Lk. 3:13; διαφορώτερον παρʼ αὐτοὺς ὄνομα, Heb. 1:4; add, 3:3; 9:23; 11:4; 12:24; see exx. fr. Grk. auth. in W. § 35, 2 b. [and as above]. ἐλαττοῦν τινα παρά τ., to make one inferior to another, Heb. 2:7, 9.
Thayer, Joseph Henry. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. New York: American Book, 1889.
Gesenius, Heinrich Friedrich Wilhelm. Hebrew Grammar. 2nd ed. Trans. Cowley, Arthur Ernest. Oxford: Clarendon Press, 1910.