How do Catholics understand Matthew 10:37?

From the Catholic point of view, in this verse our Lord says "anyone who loves their father or mother more than me, is not worthy of me". It doesn't mean we shouldn't love our parents or children, but be careful not to put our love for them above God. It is through our love for God that we can learn how to fully love others.

Our Lord revealed to us His great love for us on the Cross, and true love will always require sacrifice and self-giving. In a way, by giving God that first place, we entrust our relationships to Him and, with His guidance, can mature and grow in love of God and neighbour.

Jesus answers this question in Mark 2:27, Matthew 22:34-40 and John 13:34

You have asked a question different in detail but identical in form to when the hypocrites and Pharisees tried to play gotcha with Jesus: which law prevails when it appears that two laws are in conflict? This is hardly a Catholic unique problem to deal with (per our exchange in comments) though most pastors have ways to help the parishoner when potential conflicts arise.

Matthew 22:36-40 (KJV)

34 But when the Pharisees had heard that he had put the Sadducees to silence, they were gathered together. 35 Then one of them, which was a lawyer, asked him a question, tempting him, and saying, 36 Master, which is the great commandment in the law? 37 Jesus said unto him, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. 38 This is the first and great commandment. 39 And the second is like unto it, Thou shalt love thy neighbour as thyself. 40 On these two commandments hang all the law and the prophets.

Jesus then takes this a step further (John 13:34)

34 A new commandment I give unto you, That ye love one another; as I have loved you, that ye also love one another.

We show God our love for Him (through our obedience to his will) when we love one another. We further show love through humility, and in our selfless service to one another. In making peace and showing a sacrificial love (agape) for the parents in this case (who doubtless understand the obligations of worship but are calling on their child to serve the family interest) the child demonstrates the kind of love Jesus called on his disciples to have for one another, and in following his teaching showing love for him.

If the child is Catholic, it is best to seek pastoral counselling on how to best fulfill the Sabbath obligations from their pastor, if this situation is causing conflict within the family. Each situation likely has its own logic, details, and interpersonal factors. Unlike the cartoon character Catholics that get depicted in many questions here, real life pastors are keenly aware that the modern world presents a myriad of challenges to the faithful, and have a variety of means to assist the faithful in dealing with those problems. Every priest I know acknowledges that sometimes, despite our best intentions and desires, we are stuck with working on Sunday (for example). The USCCB has a brief treatment on that problem here.

Not everyone has the freedom to take Sundays away from work. Some people, including medical professionals and public safety workers, must work on Sundays to keep the rest of us safe and healthy. Others must work for economic reasons beyond their control.

While a part of the answer is to attend vigil mass on Saturday night (available in most places), simply going to Mass isn't the end all and be all to observing the Sabbath. Mark 2:27 (KJV)

And he said unto them, The sabbath was made for man, and not man for the sabbath:

A core principle for observing the Sabbath is to refrain from unnecessary work (brought up in our diocese with some frequency during homilies).

If it is necessary to keep the peace in the family, then that calling to love one another, and in so doing show our love for God, is consistent with Jesus' teaching. The parishoner should try to find a suitable time for peaceful rest to further fulfill the spiritual intent of the sabbath, which is where the pastor can provide both practical and spiritual guidance.

The Sunday obligation

CCC 2180 The precept of the Church specifies the law of the Lord more precisely: "On Sundays and other holy days of obligation the faithful are bound to participate in the Mass."117 "The precept of participating in the Mass is satisfied by assistance at a Mass which is celebrated anywhere in a Catholic rite either on the holy day or on the evening of the preceding day."118

CCC 2186 Christians will also sanctify Sunday by devoting time and care to their families and relatives, often difficult to do on other days of the week.

The problem with cherry picking scripture

On the other hand, when trying to understand Mattheew 10:37, it might be better to view it in the context of the surrounding verses;

32. "Whosoever therefore shall confess me before men, him will I confess also before my Father which is in heaven. 33. But whosoever shall deny me before men, him will I also deny before my Father which is in heaven." 34. "Think not that I am come to send peace on earth: I came not to send peace, but a sword. 35. For I am come to set a man at variance against his father, and the daughter against her mother, and the daughter in law against her mother in law. 36. And a man's foes shall be they of his own household." 37. "He that loveth father or mother more than me is not worthy of me: and he that loveth son or daughter more than me is not worthy of me. 38. And he that taketh not his cross, and followeth after me, is not worthy of me. 39. He that findeth his life shall lose it: and he that loseth his life for my sake shall find it." 40. "He that receiveth you receiveth me, and he that receiveth me receiveth him that sent me. 41. He that receiveth a prophet in the name of a prophet shall receive a prophet's reward; and he that receiveth a righteous man in the name of a righteous man shall receive a righteous man's reward. 42. And whosoever shall give to drink unto one of these little ones a cup of cold water only in the name of a disciple, verily I say unto you, he shall in no wise lose his reward."

The risk of cherry picking scripture is that errors in understanding are a common result. (Happens to me sometimes ...) This mornings homily was on this passage, and our pastor presented this, and Mark 9: 42-47, as Jesus making a point by using overstatement.

42 And whosoever shall offend one of these little ones that believe in me, it is better for him that a millstone were hanged about his neck, and he were cast into the sea.

43 And if your hand offend you, cut it off: it is better for you to enter into life maimed, than having two hands to go into hell, into the fire that never shall be quenched:

44 Where the worm dies not, and the fire is not quenched.

45 And if your foot offends you, cut it off: it is better for you to enter lame into life, than having two feet to be cast into hell, into the fire that never shall be quenched:

46 Where the worm dies not, and the fire is not quenched.

47 And if your eye offend you, pluck it out: it is better for you to enter into the kingdom of God with one eye, than having two eyes to be cast into hell fire: