How one approaches or accepts death is one clear psychoanalytic element in the poem. Thomas mentions four "case studies" if you will: "wise men", "good men", "wild men" and "grave men" and how they choose to face the end of their life. Then, the last man mentioned, the speaker's father, speaks to the cultural element that your question asks about. This poem is a heartfelt response to the actual situation the poet is in; his father's deteriorating health was at the forefront of his emotions. Thomas has also stated the his poetry is "a record of his own individual struggle from darkness towards some measure of light." This speaks to the psychoanalytic element as well. There are audio recordings of this poem on the Internet, with Thomas reading. His voice changes dramatically at the end of the poem, as if the fire is literally going out as he reads (when he gets to the last stanza about his father.)
The poem by Dylan Thomas was probably written in reaction to his father's serious illness. Thomas encourages the reader (or listener) of the poem to relish life for all its worth and fight against dying. He urges the reader to avoid the temptation to give up and accept death. Thomas wanted the reader to fight to the very end and to hang onto life. Looking at this with an eye toward psychoanalysis, Thomas probably thought that urging someone to fight against death would prolong that person's life. He saw his own father dying and he didn't want his father to die. Most likely, Thomas felt that the fight to avoid death would envigorate a person enough to prolong life. He eschews the romantic notion of an easy, "gentle" death.