18:9-14 Does this parable teach that Christians must close their eyes and bow their heads when they pray?

No! Six times in scripture Jesus Himself is recorded praying with His eyes open looking up to Heaven (cp Mt 14:19; Mk 6:41; 7:34; Lk 9:16; Jn 11:41; 17:1).

The parable Jesus told in Lk 18:9-14 is called the parable of the Pharisee and the publican (a publican is a tax-collector). It is not primarily about prayer, but self-righteousness and religious pride. The purpose of the parable was to expose those who trusted in their own righteousness, and despised others (cp V9).

It is about justification and getting right with God. The way to justification is expressed in the parable and summed up in what Jesus says at the end of it "…for everyone that exalteth himself shall be abased; and he that humbleth himself shall be exalted." Jesus teaches the same lesson here as he does in the parable of the labourers in the vineyard in Mt 20 (cp Mt 20:1-16). In these two parables Jesus teaches that true humility leads to exaltation.

Those who insist on their own righteousness are not only setting themselves against the purpose of God for others, but are cutting themselves off for atonement. The publican went home justified before God because he acknowledged his need for God's mercy. God acquitted him, but not the Pharisee. He came into the temple a sinner and left the same way. The Pharisee was self-righteous - he was not conscious of his sinful nature. He considered himself righteous because of his acts of piety and outward goodness.

He is justified in his own eyes and therefore does not need God's mercy. As a result the Pharisee excluded himself from God's redeeming love, and without he repented, salvation as well (cp 1Jn 1:8-10). See also comments on Mt 19:30 and 20:16.