Often, it can be out of love and protection that a parent doesn't like your boyfriend or girlfriend. They may see behavior patterns and hear turns of phrases that make them uneasy. Because our parents have more life experiences than us, they speak from their learned lessons and want to prevent us from making mistakes.
Sometimes our parents will project their unhealed trauma onto your love interest without their own awareness to your own dismay. This is a lot harder to handle as it's no one's fault, but it can be more challenging to resolve as it will take work on your parent's side and tolerance on your romantic partner's side if they are in it for the long haul.
Deep Listening
When we come from a place of deep listening, we learn how to strengthen our relationships from a place of unconditional love. Talk to your parents and sit down with them, if you can. Hear what they have to say, without judgment and without reacting to their concerns.
Listen more in-depth than what they tell you on the surface, and be prepared to ask gently probing questions like "what exactly bothers you?" or "who do they remind you of?" Answers to these questions require you to be mindfully present and nonreactive.
Be open and receptive to hear and allow them to go on until they have nothing left to offer. Only asks questions without any counter-arguments. This will help your relationship as you build trust with your parents, who might be questioning your judgment based on their own past experiences.
Honor Each Other's Feelings
After hearing them out, take some time to think over what they have told you. See if you can determine the source of their feelings. Allow yourself to sit with it for a few days. If you didn't feel like you could gain clarity from them about the source, reach out to family and friends and ask their advice. See if there are people from their past who can give you some insights into their reactions. You may even want to reach out to a psychic empath to see if they can offer you some guidance into what is going on with them.
After a few days, ask to meet with them again and hear you out. Come from a place of kindness and compassion. Let them know how what they had to say made you feel and that you listened to them and considered everything. Maybe after doing so, you decided they were right. If not, then you need to be clear about the next step.
Set Firm Boundaries
Be clear about what you want out of your relationship with your person of interest and what you expect from your parents. At this point, you are standing your ground if you've decided to stay in the relationship. If needed, spend time beforehand thinking about what dos and don'ts are acceptable from your parents. For example, do you need limits on how often they contact you? What behaviors did you expect from them when they come in contact with your boyfriend or girlfriend?
Also, be receptive to hearing what boundaries they need from you. They may ask that your person of interest not come over to their own home until they feel comfortable with them. This is an important boundary, but one that may have to change if the two of you move in together or become engaged.
Practice Patience
Your parents may need time to learn to like your romantic partner or work through whatever projections they are placing on them. Handle these conflicts gently and without resentment. Doing so will build trust with your parents, and even if they don't like your choices, they may learn to respect you better.
Speak with your Romantic Partner
Most importantly, you will need to discuss all this with your boyfriend or girlfriend. They will need to come to terms with whatever findings you realize, even if you can't tell them all the details. It may be a deal-breaker for the relationship if you don't allow them space to process and honor their feelings. Your parents may need time, but any ill behavior and lack of acceptance of your romantic partner will cause emotional harm to your love interest. Be compassionate to them as well and supportive. They need to know you have their back, especially if your parents don't like them.