I recommend that you check out the following paper in which the authors provide a very nice and easy-to-read overview of various fit statistics in CFA/SEM:

K Schermelleh-Engel, H Moosbrugger, H Müller (2003): Evaluating the fit of structural equation models: Tests of significance and descriptive goodness-of-fit measures. Methods of Psychological Research Online 8 (2), 23-74

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.509.4258&rep=rep1&type=pdf

Many fit statistics are based on the comparison of the observed covariance (and sometimes mean) structure with the model-implied (model-estimated) covariance (and potentially mean) structure. The chi-square test of model fit tests the null hypothesis that there is an exact fit between the observed covariance (and/or mean) structure and the model-implied covariance (and/or mean) structure in the population. Other fit statistics such as the SRMR and RMSEA are also based on the comparison of the observed and model-implied covariance and/or mean structures, but these indices are more "descriptive" in nature, quantifying the degree of model misfit (rather than testing a formal null hypothesis of exact fit). Yet other indices such as the CFI and TLI are based on the comparison between the target (estimated) model and a so-called null or independence model. It is common to report the chi-square test value, df, and p value (a non-significant p value indicates that the model does not have to be rejected). In addition, people frequently report RMSEA, SRMR, and CFI/TLI as descriptive or "closeness of fit" indices. For details, see the above paper.

It should be noted that a good model fit does not necessarily indicate that your validation was successful. Even a well-fitting model may contain parameter estimates (e.g., factor loadings, factor correlations, etc.) that are not in line with your theoretical expectations, for example, with regard to convergent and discriminant validity. Assessment of overall model fit is only the first step in model evaluation. You also want to take a close look at the parameter estimates, their standard errors, etc.

The usual criteria discussed by Hu and Bentler (1998 and 1999) apply. I generally use the chi-square test for departure from fit, CFI and SRMR plus RMSEA where there are latent variables (i.e. factors). In that circumstance, an additional consideration applies: Am I assuming the correct number of latent variables?

That determination is considered the most critical in exploratory factor analysis-- much more important than the chosen methods for factor extraction and factor rotation. It is generally accepted that parallel analysis is the soundest method of determining how many factors to extract. I see no reason why that determination should be less important in the context of confirmatory factor analysis. So parallel analysis is where I'd start.

The Hu and Bentler (1999) criteria for "close" or "approximate" fit mentioned by Paul Max Kohn are widely used. However, they are not without problems as shown by, for example,

Article In Search of Golden Rules: Comment on Hypothesis-Testing App...

McNeish, D., & Wolf, M. G. (2021). Dynamic fit index cutoffs for confirmatory factor analysis models. Psychological Methods. Advance online publication. https://doi.org/10.1037/met0000425.

As indicated by these and other studies, the criteria for indices such as RMSEA, SRMR, and CFI/TLI generalize poorly across different conditions and are therefore essentially worthless in practice.

I don't think that parallel analysis is needed when conducting confirmatory factor analysis (CFA). CFA requires you to have an a priori theory about the number of factors and the loading pattern. In other words, CFA does not explore the number of factors. The formal test of fit (chi-square test) is a very powerful test that tends to reliably indicate when there is an insufficient number of factors considered in CFA or when there are other sources of misfit/misspecification. Parallel analysis is not needed for that purpose, and, more importantly, it may be misleading when the factor structure is complex with variables loading onto more than one factor (e.g., when there are method factors in addition to substantive factors).

A priori models are all very well if they correspond to reality. Otherwise, not so much. How can they correspond to reality if based on an incorrect number of factors? One needs to remember that the composition of the first three factors will differ between a three-factor model and a six -factor model based on the same data. In other words, the number of factors will affect the interpretation of all factors in a solution.

Paul Max Kohn I'm not saying that a model with an incorrect number of factors corresponds to reality or that it should be selected/accepted/used. What I'm saying is that parallel analysis is not needed in CFA to reject an incorrect model. The chi-square test of model fit is highly sensitive to underspecified models that contain too few factors or other specification error that leads to statistically significant covariance and/or mean residuals.

Bentler and Bonett (1980) pointed out that the chi-square test is hypersensitive to sample size. In the case of confirmatory factor analysis of questionnaire responses, I think that means people rather than items. Also, one should avoid overspecified models with too many factors as well as underspecified ones.

As well, it's good to have an idea of how many factors there are even before conceptualizing a model. I'd even go so far as to conduct exploratory factor analysis before confirmatory factor analysis. Some might view this as cheating, but I see it as common sense in the pursuit of the common goal of advancing real knowledge and avoiding detours in the pursuit.

Any test of significance is "sensitive" to sample size. This isn't problematic but a desirable feature (increased power to detect model misspecification). It means that as your sample size goes up you will find it easier to reject an incorrect model. Closeness-of-fit (approximate) indices such as RMSEA, CFI etc. do not share this feature. As the sample size increases, you are less likely to detect misspecification with these indices which is paradoxical. See

Article In Search of Golden Rules: Comment on Hypothesis-Testing App...

As an aside, the chi square (and associated approximate fit statistics) does have sensitivity to number of items (model size), which is often ignored (model size effect). See, for example:

Article Revisiting the Model Size Effect in Structural Equation Modeling

Article Understanding the Model Size Effect on SEM Fit Indices

There are correction formulas for this effect that appear to work fairly well in practice.

All of the pro and con arguments with regard to chi square vs. approximate fit statistics and also with regard to the "EFA before CFA debate" have been laid out in great detail on the SEMNET listserv and can be found in the archives at https://listserv.ua.edu/cgi-bin/wa?REPORT=SEMNET&z=4&L=SEMNET&1=SEMNET&X=&Y=

According Hu & Bentler (1995): For RMSEA the criteria is: good fit: 0.10); and for TLI, IFI and CFI: good fit: > 0.95; acceptable: 0.90 to 0.95; poor fit: < 0.90.

According Chen, 2007: The chi-square, and change in the RMSEA and CFI values can be used for examining differences in model fit. The invariance criteria is ΔCFI≤ 0.01, and ΔRMSEA ≤ 0.015.