Adherents to this line of the Westminster Standards who nonetheless reject exclusive psalmody argue in three primary ways:

• The Westminster divines were divided on the issue
• The Confession does not specify that only psalms may be sung
• The word "psalms" should not be understood as necessarily referring only to the Book of Psalms

Division at Westminster

J. V. Fesko, The Theology of the Westminster Standards, 357ff., points to evidence that at least some of the Westminster divines rejected exclusive psalmody, and would thus have objected to language in the Confession mandating it. Two of the passages that the Assembly cited in defense of the "singing of psalms" clause are Colossians 3:16 and Ephesians 5:19, which refer to "psalms, and hymns, and spiritual songs." These verses were interpreted differently by the two groups: exclusive psalmodists argued that this refers to three divisions of the Book of Psalms, while others saw it as referring first to the Psalms and then to two other types of music.

He quotes Edward Leigh:

The Apostle speaketh of Psalmes, Hymnes and spiritual Songs, Ephes. 5. 19. and Col. 3. 16. Who can shew any reason to limit his speech to Scripture-psalms? Why may not one praise God in a Song for our deliverance in 88, or the Gun-pouder treason? (source)

Similarly, the Annotations of Westminster on Ephesians 5:19 read:

But it seemeth most probable that by Psalms he meaneth the Psalms of David—set to the harp or psaltery—by Hymns, certain ditties made upon special occasions—and by spiritual songs, such as were not composed beforehand, and pricked before them with musical notes, but such as men edited by an extraordinary gift. (source)

It would seem, then, that the Confession may have been written in such a way as to permit these views. Analysis of the Assmebly's proceedings is not possible, as they have been lost, so we must settle the question in other ways. Those who say that the Assembly did not intend to prohibit exclusive psalmody continue the argument in two primary ways.

Non-psalms not prohibited

John Gerstner sums up the first argument:

[The Westminster Confession] mentions singing of psalms. It doesn't say psalms exclusively, though the tendency at that time was to sing psalms exclusively. (source; 22:50)

Westminster's Directory of Public Worship goes into more detail but again refers only to psalms:

It is the duty of Christians to praise God publickly, by singing of psalms together in the congregation, and also privately in the family. [...] (source)

Fesko concludes:

The DPW states only that the psalms should be sung, not that they should be sung exclusively. The DPW is silent regarding the use of non-inspired songs in worship.

"Psalms" includes hymns

For the next argument, Robert Rayburn refers to the lack of mention of "hymns and spiritual" songs in the Westminster Confession:

This omission does not mean that we should sing the Old Testament psalms only. The Confession uses the word in a wider sense to refer to hymns sung to God. (O Come, Let Us Worship, 95)

That is, the argument goes, the word "psalms" in the Confession would have been understood to include more than the Book of Psalms. Fesko summarizes the work of Nick Needham:

Recent research surveying the views of a number of sixteenth- and seventeenth-century theologians, such as John Daillie (1584–1670), Matthew Poole (ca. 1624–1679), and Thomas Manton (1620–1677), confirms the broad use of the term psalm to denote scriptural and extra-scriptural religious musical compositions.

For example, Poole, citing Luke 24:44, indicates that understanding "psalms" as a general term was not unheard of:

[Others take psalms] more generally, as the genus, noting any holy metre, whether composed by the prophets of old, or others since, assisted by the Spirit extraordinarily or ordinarily. (source)

Conclusion

Fesko concludes:

the most likely scenario is that the Standards promote the inclusive use of psalmody in worship as a necessary element but are silent regarding the use of non-inspired scriptural songs in worship.

Incidentally, both denominations mentioned in the question encourage the singing of psalms in worship, but not to the exclusion of appropriate hymns:

Congregations do well to sing the metrical versions or other musical settings of the Psalms frequently in public worship. Congregations also do well to sing hymns of praise that respond to the full scope of divine revelation. (OPC DPW II, B.2.c; cf. PCA BCO 51-3)