The ability to believe the Gospel -- that is, deposit of faith -- is given everyone, or Christ would not be calling us to have childlike faith. The distinction with the parables is temporal, preceding the Resurrection and the Great Commission, according to which the entire gospel is to be preached to "all creation". The church is sent to overcome the limited ability of some to understand the scripture; such understanding, however, cannot be equated with faith.
I find it, however, remarkable that you acknowledge that the call is to all. Is it the standard Calvinist teaching? How does it square with the notion that God does not want all to be saved?
1Co 2:14 But a natural man does not accept the things of the Spirit of God, for they are foolishness to him; and he cannot understand them, because they are spiritually appraised.
2Th 3:2 and that we will be rescued from perverse and evil men; for not all have faith.
Of Effectual Calling
I. All those whom God hath predestinated unto life, and those only, He is pleased, in His appointed time, effectually to call, by His Word and Spirit, out of that state of sin and death, in which they are by nature to grace and salvation, by Jesus Christ; enlightening their minds spiritually and savingly to understand the things of God, taking away their heart of stone, and giving unto them an heart of flesh; renewing their wills, and, by His almighty power, determining them to that which is good, and effectually drawing them to Jesus Christ: yet so, as they come most freely, being made willing by His grace.
II. This effectual call is of God's free and special grace alone, not from anything at all foreseen in man, who is altogether passive therein, until, being quickened and renewed by the Holy Spirit, he is thereby enabled to answer this call, and to embrace the grace offered and conveyed in it.
III. Elect infants, dying in infancy, are regenerated, and saved by Christ, through the Spirit, who works when, and where, and how He pleases: so also are all other elect persons who are incapable of being outwardly called by the ministry of the Word.
IV. Others, not elected, although they may be called by the ministry of the Word, and may have some common operations of the Spirit, yet they never truly come unto Christ, and therefore cannot be saved: much less can men, not professing the Christian religion, be saved in any other way whatsoever, be they never so diligent to frame their lives according to the light of nature, and the laws of that religion they do profess. And to assert and maintain that they may, is very pernicious, and to be detested. Check here for the Westminster Confession scriptural references