Am I not allowed to grieve when a loved one dies?

First of all, grief is biblical (cf. Eccl. 7:2; Psa. 30:5; 56:8). It is a sign of deep love and affection. An important aspect of dealing with our sorrows is the expression of our grief before our Father (1 Pet. 5:7; cf. 1 Sam. 12:22-23). It’s also important to share it with others (Gal. 6:2). The death of Lazarus is a good example (John 11:31-44).

But let's look at the text you cited:

1 Thessalonians 4:13-18: But we do not want you to be uninformed, brothers, about those who are asleep, that you may not grieve as others do who have no hope. For since we believe that Jesus died and rose again, even so, through Jesus, God will bring with him those who have fallen asleep. For this we declare to you by a word from the Lord,[a] that we who are alive, who are left until the coming of the Lord, will not precede those who have fallen asleep. For the Lord himself will descend from heaven with a cry of command, with the voice of an archangel, and with the sound of the trumpet of God. And the dead in Christ will rise first. Then we who are alive, who are left, will be caught up together with them in the clouds to meet the Lord in the air, and so we will always be with the Lord. Therefore encourage one another with these words.

From this text we observe that the apostle Paul is actually teaching that Christians are to grieve when a Christian loved one passes on into glory (cf. Rom. 12:15). However, we are to grieve differently. We are to grieve "in hope," because we will see them again. We will be separated from our Christian loved ones only for a brief period of time. All our pain and sorrow will ultimately be resolved on the last day when our glorious Lord returns. Christians will dwell with each other for eternity in the new heavens and new earth. So, we are to grieve in hope because we’re being comforted with the truth that Jesus is returning and with him all the saints that rose earlier. Our grief is only a temporary condition.